USFDC Home  USF Electronic Theses and Dissertations   RSS 
Material Information
Subjects
Notes
Record Information

Full Text 
PAGE 1 Nu merical Simulations of Heat Transfer Processes in a Dehumidifying Wavy Fin and a Confined Liquid Jet Impingement on Various Surfaces by Mutasim Mohamed Sarour Elsheikh A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Muhammad M. Rahman, Ph.D. Frank Pyrtle, III, Ph.D. Stuart Wilkinson Ph.D. Date of Approval: March 23 2011 Keywords: Fully Confined Fluid Jet Impingement, Steady State, Transient Analysis, Conjugates Heat Transfer, Heat Flux Copyright 2011, Mutasim Mohamed Sarour Elsheikh PAGE 2 Dedication To God In the name of Allah, Most Gracious, Most Merciful I bear witness that there is no God but Allah, and Mohamed is the Messenger of Allah. To my mother, Azeeza t hank you for everything that you have done, for all of your love and support I also wish to thank my wife, Haifa, and our lovely kids, Azeeza and Magdi and my d ear brother Magdi I dedicate this thesis especially in memory of my father, Mohamed Sarour Fin ally, to my Major Professor Muhammad Mustafizur Rahman, t hank you for your e normous tolerance and guidance. W ithout your guidance and persistent help this thesis would not have been materialized PAGE 3 Acknowledgement s Apart from individual effort the s uccess of any project depends largely on the encouragement and guidance of many others. I take this opportunity to express my gratitude to the people who have been instrumental in the successful completion of this project. I would like to express my since re appreciation to Mr. Bernard Batson for his support and encouragement. I would like to also thank Donovan Industries (Tampa, FL) and the generous support from the National Science Foundation S STEM grant DUE #0807023. Also, I would like to thank Professor Frank Pyrtle and Professor Stuart Wilkinson for being my committee members. I am grateful for their constant support and help. Special t hanks to Dr. Luis Rosario and Dr. Jorge C. Lallave, for their guidance and valuable support. I n addition, I sincerely thank Professor Ashok Kumar for his help and guidance. I wish to thank John Shelton and Daniel Miller for their help in r evising my thesis. Last but not least, I would like to thank Sue Britten, Shirley Tervort, and Wes Frusher for their help. PAGE 4 i Table of Contents List of Tables ...................................................................................................................... i i List of Figures .................................................................................................................... iii List of Symbols .................................................................................................................. vi Abstract ................................................................................................................................x Chapter 1: Introduction and Literature Review ...................................................................1 1.1 Introduction (Heat Transfer in a Wavy Fin Assembly ) .....................................1 1.2 Literature Review (Heat Transfer in a Wavy Fin Assembly ) ............................2 1.3 Introduction (Heat Transfer by Jet Impingement ) .............................................6 1.4 Literature Review (Heat Transfer by J et Impingement ) ....................................7 Chapter 2: Heat Transfer Analysis of Wav y Fin Assembly with Dehumidification .........12 2.1 Physical Description of Wavy Fin Heat Exchanger s .......................................12 2.2 Mathematical Model ........................................................................................18 2.3 Results and Dis cussion ....................................................................................23 Chapter 3 : Conjugate Heat Transfer Analysis of a Confined Liquid Jet Impingement on Concave and Convex Surfaces .............................................36 3.1 Modeling and Simulation .................................................................................36 3.2 Results and Discus sion ....................................................................................41 Chapter 4: Conclusions ......................................................................................................62 References ..........................................................................................................................65 Appendices .........................................................................................................................70 Appendix A: Q Basic Heat Transfer Code of a Wavy Fin Analysis .....................71 Appendix B: FIDAP Code for Analysis of Heat Transfer by Jet Impingement ....................................................................................................79 PAGE 5 ii List of Tables Table 2 .1 Geometr ic dimensions of sample wavy fin and tube heat exchangers 16 Table 2 .2 Geomet ric dimensions of sample fin and tube heat exchangers 1 7 PAGE 6 iii L ist of Figures Figure 2.1 Most evaporator uses in air condition systems 12 Figure 2.2 Schematic diagram of evaporator 13 Figure 2. 3 Some types of fins. 13 Figure 2. 4 Side views of a wavy fin assembly 14 Figure 2. 5 Side views of the physical wavy model 14 Figure 2. 6 Side views of the physical street radial model 15 Figure 2. 7 Variation of dimensionless temperature distribution with the variation in relative humidity. 24 Figure 2. 8 Variation of dimensionless temperature distribution with variation in cold fluid temperature at relative humidity 50%. 25 Figure 2. 9 Variation of dimensionless temperature distribution with variation in surrounding air dry bulb temperature at relative humidity 50%. 27 Figure 2. 10 The present model with and without insulation in the fin tip, and at 50% RH. 28 Figure 2. 11 Comparison of rectangular and wavy models for dry and 50% RH. 29 Figure 2.12 Comparis on between present model and Kazeminejad [6] ; and Rosario and Rahman [10] 30 Figure 2.13 (Aug)dry/(Aug)wet variation with change in T1. 31 Figure 2.14 (Aug)dry/(Aug)wet variation with change in T2. 32 Figure 2.15 (Aug)dry/(Aug)wet variation with change in RH 33 PAGE 7 iv Figure 2.16 Compari son of 1 D and 2 D radial models for dry and 50% RH 34 Figure 2.17 Comparison of the wavy model and the converted rectangular model at dry and 50% RH. 35 Figure 3.1 Two dimens ional liqu id jet impingement on a uniformly heated concave surface. 36 Figure 3.2 Velocity vector distribution for jet imping e ment on a curve d copper plate. 41 Figure 3.3 Solid fluid interface temperature for different number of elements in x and y directions (Re = 1,000, b = 30, w = 0.6 cm) 42 Figure 3. 4 Dimensionless interface temperature and Local Nusselt number distribution for (a) concave and (b) convex copper pla te at different Reynolds number s and water as the cooling f1uid. 44 Figure 3.5 A verage Nusselt number at different Reynolds number s for (a) concave (b) co nvex copper plate with water as the cooling fluid (R= 6 .21, 6.61, 7.01, and c m) 46 Figure 3.6 S olid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for a concave copper wafer at different r adius, and water as the cooling f1uid (R= 6 .2 1, 6.61, 7.01, and c m) 48 Figure 3.7 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for a convex co pper wafer at different r adius, and water as the cooling f1uid (R= 6 .2 1, 6.61, 7.01, and c m) 49 Figure 3.8 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different material thickness (H=1, 1.5, 2, 2.5, and 3 cm). 51 Figure 3.9 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different spacing of concave curvature (D=0.1, 0.2, 0.3, 0.4, and 0.5 cm) 53 PAGE 8 v Figure 3.10 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) L ocal Nusselt number distribution for different spacing of convex curvature (D = 0.1, 0.2, 0.3, 0.4, and 0.5 cm). 54 Figure 3.11 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different materials ( c opper, silicon, a luminum, and Constantan). 56 Figure 3.12 Dimensionless interface temperature and Local Nusselt number distribution for (a) concave and (b) convex s ilicon plate s at different Reynolds number s and water as the cooling f1uid. 59 Figure 3.13 Stagnation Nusselt number comparison of Rahman et al. [49 ], Inoue et al. [4 0 ], with actual numerical results under different Reynolds numbers ( w = 4 mm, d = 2 mm). 61 PAGE 9 vi List of Symbols b Inner solid thickness [m] Bi Fin Biot number, h2t/kf Bi1 Cold fluid Biot number, h1p/kw Bi2 Ambient Biot number, h2p/kw cpa Specific heat of dry air [J/kg oC] d Channel spacing [m] h Heat transfer coefficient [W/m2K], qint/(TintTj) h1 Cold fluid heat transfer coefficient [W/m2oC] h2 Ambient heat transfer coefficient [W/m2oC] hd Mass transfer coefficient [kg/m2s] hm Mass transfer coefficient [J/kg] hn Height of the nozzle from the plate [m] hfg Latent heat of condensation [J/kg] H Outer solid thickness k T hermal conductivity [W/m K] kf fin thermal conductivity [W/moC] kw Wall thermal conductivity [W/moC] K Thermal conductivity ratio (fin wall), (kf/kw) Mass flow rate of air [kg/s] PAGE 10 vii Nu Nusselt number, (hdn) / kf Nuav Average Nusselt number for the entire surface, (havdn)/kf pt Half fin pitch [m] p Pressure [Pa] P Aspect ratio, (t/p)q Heat flux [W/m2] qc latent heat flux [W] qs Sensible heat flux [W] qt Total heat flux, qs + qc [W] r Radial coordinate [m] R Ratio of sensible to total heat transfer calculated at fin temperature Rb Ratio of sensible to total heat transfer rate at fin base temperature R1 Ratio of sensible to total heat transfer rate at fin tip temperature R0 Outer radius of curvature [m] Ri I nner radius of curvature [m] Re Reynolds number, (Vjw)/ f RH Relative humidity RT Fin tip radius [m] t Half fin thickness [m] T Temperature [oC] T1 Cold fluid temperature [oC] T2 Ambient dry bulb temperature [oC] Vj Jet velocity [m/s] w Wall thickness [ m] PAGE 11 viii wn Nozzle width [m] Humidity ratio at the surface Humidity ratio at any point W Dimensionless wall thickness (w)/p x x coordinate [m] Greek Symbols Dimensionless fin thickness ( o/p) ma Mosit air density [kg/m3] 1 Fin angle [ Fin efficiency 1 Dimensionless temperature (T T2)/(T1T2) Thermal diffusivity [m2 /s] Channel spacing ratio, b/d Thermal conductivity ratio, ks/kf Angular coordinate [rad] Solid thickness to curvature ratio, b/RO Kinematic viscosity [m2/s] Angular coordinate [rad] Dimensionless temperature, 2 kf (Tint Tj) / (q w) Density [kg/m3] Radius of curvature, RO / d Subscripts PAGE 12 ix 1 Cold fluid side 2 Air side avg Average e Outer edge of the fin f Fin f Fluid i Interfacial location int Solid fluid interface j Jet or inlet max Maximum n Nozzle s Solid w Wall Condensate film PAGE 13 x Abstract This thesis consists of two different research problems. In the first one, the heat transfer characteristic of wavy fin assembly with dehumidification is carried out. In general, f in tube heat exchangers are employed in a wide variety of engineering applications such as cooling coils for air conditioning, air pre heaters in power plants and for heat dissipation from engine coolants in automobile radiators. In the se heat exchangers a heat transfer fluid such as water, oil, or refrigerant, flow s through a parallel tube bank, while a second heat transfer fluid s uch as air is directed across the tubes. Since the principal resistance is much greater on the air side than on the tube side, enhanced surfaces in the form of wavy fins are used in air cooled heat exchangers to improve the overall heat transfer performan ce. In heating, ventilation, and air conditioning systems (HVAC) t he air stream is cooled and dehumidified as it passes through the cooling coils circulating the refrigerant. Heat and mass transfer take place when the coil surface temperature in most coo ling coils is below the dew point temperature of the air being cooled. This thesis presents a simplified analysis of combined heat and mass transfer in wavy finned cooling coils by considering condensing water film resistance for a fully wet fin in dehumid ifier coil operation during air condition. The effects of variation of the cold fluid temperature ( 5 5 temperature (25 35 ) and relative humidity (50% 70%) on the dimensionless temperature distribution and the augmentation factor are investigated and compared with PAGE 14 xi those under dry conditions. In addition, comparison of the wavy fin with straight radial or rectangular fin under the same conditions were investigated and the results show that the wavy fin has better heat dissipation be cause of the greater area. The result s demonstrate that the overall fin efficiency is dependent on the relative humidity of the surrounding air and the total surface area of the fin. In addition, the findings of the present work are in good agreement with experimental data. T he second problem investigated is the heat transfer analysis of confined liquid jet impingement on various surfaces. The objective of this computational study is to characterize the convective heat transfer of a confined liquid jet impinging on a curved surface of a solid body, while the body is being supplied with a uniform heat flux at its opposite flat surface. Both convex and concave configurations of the curved surface are investigated. The confinement plate has the same shape as the curved surface. Calculations were done for various solid mater ials, namely copper, aluminum, C onstantan, and silicon; at two dimensional jet. For this research, Reynolds number s ranging from 750 to 2000 for various nozzle widths channel spacing, radi i of curvature, and base thicknesses of the solid body were used Results are presented in terms of dimensionless solid fluid interface temperature, heat transfer coefficient, and local and average Nusselt numbers. The increment s of Reynolds numbers incre ase local Nusselt number s over the entire solidfluid interface. Decreasing the nozzle width channel spacing plate thickness or curved surface radius of curvature all enhanced the local Nusselt number Results show that a convex surface is more effective compar ed to a flat or concave surface. Numerical simulation results are validated by comparing them with experi mental data for flat and concave surfaces. PAGE 15 1 Chapter 1: Introduction and Literature Review 1.1 Introduc tion (Heat Transfer in a Wavy F in A ssembly) In traditional refrigeration and air conditioning systems, finned tube heat exchangers are used to cool and dehumidify air. An air stream is cooled and dehumidified by the refrigerant that is circulating through the coil tube. The evaporation of the refrigerant within the coil removes heat from the air stream. The efficiency of the fin attached to the outer surface of the coil tube is directly related to the effectiveness of the heat exchanger. The cooling process occurs by the removal of sensible heat followed by condensation of water vapor contained within the air, as the moist air passes through the coil. Simultaneously, a condensation process entails heat transfer with phase change and the cooling takes place by the removal of sensible as well as latent heat. An important quantity that controls the heat transfer rate during a dehumidification process is the ratio of sensible to total heat transfer, which is mostly used in sizing cooling coils for air con ditioning units. The current work is carried out through a one dimensional analysis and modeling of a wavy fin as used in a cooling coil (dehumidifier) of an air conditioner. The focus of the analysis is on the fully wet condition. Since the coil surface temperature in most cooling coils is below the dew point temperature of the air being cooled, simultaneous heat and mass transfer takes place. Moisture condensation on the fin surface affects the overall fin efficiency. In an air conditioner, the c ooling coils are used for the removal of PAGE 16 2 heat and moisture from the occupied space Condensation of moist air burst s onto these cooling coils located within the air conditioning units. The metal fin attached to the tube improves the heat conduction. A number of physical parameters affect the thermal performance of the cooling c oils such as geometry, material properties, psy chrometric conditions, and the efficiency of the fin. The fin efficiency may be affected when moist air is condensed on the fin. This ha ppens when the fin temperature is below that of the dew point temperature of the arriving air passing through the cooling coil. T he improvement of the efficiency of the cooling coils directly contributes to the improvement of the performance of heating vent ilation air conditioning system (HVAC), leading to big energy savings. The condensation process involves both heat and mass transfer; simultaneous cooling occurs by the removal of sensible as well as latent heat. An important quantity that used in the des ign and sizing of cooling coils for air conditioning units is the ratio of sensible to total heat transfer. 1.2 Literature Review (Heat Transfer in a Wavy Fin Assembly) Lunardini and Aziz [1] presented a review of the analytical and experimental progre ss made in understanding the process of condensation on extended surfaces. They discussed the topic of dehumidification of air on finned cooling coils. Their review is focused on rectangular fins. They reviewed models based on classical fin theory for dry fin introducing some modifications to take into account the effect of mass transfer. They concluded that although progress has been made in understanding condensation of cooling coils more theoretical and experimental works are needed. Experimental data for the overall performance of dry and fully wet cooling coil s with dehumidification have PAGE 17 3 been reported by various investigators (Kays and London [2], Wang et al. [3], Leu et al. [4]). These studies confirmed that the performance of cooling finned coils is significantly reduced when condensation takes place. This is a consequence of lower fin efficiency for wet conditions. It was shown that fully wet fin efficiency was lowe r than that of dry fin. However, only a few theoretical works have been reported on condensation assuming fully wet fins or fin assemblies (Webb [5]). Kazeminejad [6] presented a simple model for simultaneous heat and mass transfer to a cooling and dehumidifying rectangular fin. He showed an analysis of rectangular one dimensional fin assembly heat transfer with dehumidification under fully wet conditions, incorporating the ratio of sensible to total heat transfer. Salah El Din [7] presented an analytical s olution for the performance of partially wet rectangular fin assembly. His model was useful in prediction of wet and dry parts of the fin assembly beside s the effect of the various parameters including the assembly dimensions on the thermal performance. However, most dehumidifier cooling coils have annular fins in contrast to rectangular fins. Liang et al. [8] examined the efficiency of a platefin tube heat exchanger using 1D and 2 D models. The 2D model considered the complex fin geometry and the var iation of the moist air properties over the fin. Ros ario and Rahman [9] presented a one dimensional radial fin assembly model with condensation. Their findings indicated that the heat transfer rate increased in increment s in both dry bulb temperature and r elative humidity of the air. Rosario and Rahman [10] presented the 1D analysis of heat transfer in a partially wet circular fin assembly during dehumidification. These models assumed that droplets can drain off the fin under the influence of the gravitati onal force neglecting the thermal resistance of the condensate. PAGE 18 4 Threlkeld [11] proposed a rectangular fin model assuming that the fin was covered with a uniform condensate film. He developed an analytical expression for the overall fin efficiency by using the enthalpy difference as the driving potential for simultaneous heat and mass transfer. He assumed a linear relationship between the ambient air temperature and the corresponding saturated air temperature. His model showed that the wet fin efficiency was only slightly affected by the air relative humidity. ARI Standard 41081 [12] used an approach similar to Threlkeld [11], but neglecting the presence of the water film on the fin surface. McQuiston [13] developed an expression for wet fin efficiency for the case of a plane fin by approximating the saturation curve on the psychrometric chart by a straight line over small range of temperatures. Coney et al. [14] presented a numerical solution for condensation over a rectangular fin, taking into account the thermal resistance of the condensate film and using a second degree polynomial to relate the humidity ratio with dry bulb temperature. He assumed a linear temperature profil e for the condensate film. The results showed that there is negligible effect of c ondensate thermal resistance on the fin temperature distribution. Srinivasan and Shah [15] presented a summary of previous studies on condensation over rectangular fins. Elmahdy and Biggs [16] obtained the overall fin efficiency of a circular fin by taking into consideration the temperature distribution over the fin surface. Their work treated heat transfer and mass transfer separately by considering their respective driving force and then assumed a linear relationship between the humidity ratio of the saturated air on the fin surface and its temperature. Their numerical results indicate that the fin efficiency strongly depends on the relative humidity. As the relative humidity of air PAGE 19 5 increases, the driving potent ial for mass transfer increases; this leads to a higher latent heat transfer and higher temperature. McQuiston and Parker [17] presented an analysis of circular fins using an approximation proposed by Schmidt [18]. Their model assumed a linear relationship between the humidity ratio and the dry bulb temperature. Hong and Webb [19] derived an analytical formulation of fin efficiency of fully wet surface for circular fins. Their formulation was based on the exact solution of the governing differential equation after incorporating a linear relationship between the humidity ratio and the dry bulb temperature (McQuiston [13], McQuiston and Par ker [17]). Wang et al. [3] derived a fully wet fin efficiency for circular fins using the formulation given by Threlkeld [11]. They obtained an analytical expression for the fully wet fin efficiency by utilizing the enthalpy difference as the driving force for the combined heat and mass transfer process. Lin et al. [20] presented an experimental study on the performance of a rectangular fin in both dry and wet conditions. They observed that the dehumidification phenomenon can be classified into four regions. One of those regions had a thin film of condensate. Heggs and Ooi [21] presented a mathematical model for a radial rectangular fin. They presented charts that can b e used to rate or design specific radial rectangular fins for a particular heat transfer specification. However, their model did not take into account any condensate effect. Lin and Jang [22] presented a 2 D analysis for the efficiency of an elliptic fin under the dry, partially wet and fully wet conditions for a range of axis ratios. One limiting condition was the circular fin. The objective of the present work is to develop an analytical solution for heat and mass transfer in a wavy fin assembly under w et conditions, considering that the fin is covered with a uniform condensate film. This analysis also studies the effects of variation PAGE 20 6 of cold fluid temperature ( 5 5 (25 35 humidity (50 70 ) on the di mensionless temperature distribution and the augmentation factor compared with those under dry condition. The results are expected to be mean ing ful for the design of cooling coils for air conditioning. 1.3 Introduction (Heat Transfer by Jet Impingement) There are numerous experi mental and theoretical studies on the characteristic s and heat transfer associated with confined jet impingement on surfaces. These studies have considered both single i mpinging jet and jet arrays. Martin [ 23] and Viskanta [24] reviewed earlier studies of impingement heat transfer Jet impingement has been demonstrated to be an effective means of providing high heat/mass transfer rates in industrial processes where rapid heating, cool ing, or drying is necessary These include surface coating and cleaning, cooling of electronic components fi re testing of building material, annealing of metal and plastic sheets tempering glass, chemical vapor deposition, avionics cooling cooling of turbine blades, and drying of textiles, according to Hong et al. [25 ]. The principal virtue of this method of cooling is the large rate of heat transfer and the relative ease with which both the heat transfer rate and distribution can be controlled. T here are only a few studies on concave and convex surfaces while several studies of impinging jets are for flat surfaces If the fluid is discharged from a nozzle or orifice into a body of surrounding fluid that is the same as the jet itself, then it is c alled submerged. Confined submerged li quid jets find use in both axi symmetric and planar configurations Both configurations share the common feature of a small stagnation zone PAGE 21 7 at the impingement surface whose size is of the order of the nozzle diameter or slot dimension, with the subsequent formation of a wall jet region. The fluid impingement and boundary layer behaviors that control the convective heat transfer will be examined for two dimensional under confinement conditions in the present investigation. 1.4 Literature Review (Heat Transfer by Jet Impingement) The following is a summary of most related literature pertaining to confined and semi confined jet impingement over flat, concave, and convex surfaces. Glauert [26 ] considered the flow due to jet spreading out over a plane surf ace, either radial ly or in two dimensions. Solutions to the boundary layer equations were sought for a laminar flow using similarity transformation. McMurray et al. [ 27] studied convective heat transfer to an impinging plane jet from a uniform heat flux wall. To fit their data, they based heat transfer correlations on the stagnation flow in the impingement zone and on the flat plate boundary layer in the uniform parallel flow zone. Metzger et al. [ 28] experimentally studied the effects of Prandtl number on heat transfer to a liquid jet for a uniform surface temperature boundary condition. Thomas et al. [ 29] measured the film thickness across a stationary and rotating horizontal disk using the capacitance technique, where the li quid was delivered to the disk by a controlled impinging jet. Faghri and Rahman [ 30] experimentally, analytically, and numerically studied the heat transfer effect from a heated stationary or rotating horizontal disk to a liquid film from a controlled impi nging jet under partially confined conditions for different volumetric flow rates and inlet temperatures for both supercritical and subcritical regions. Hung and Lin [ 31] proposed an axi symmetric sub channel model for evaluating local surface heat flux fo r confined PAGE 22 8 and unconfined cases. Garimella and Rice [32] presented experimental results for the distribution of local heat transfer coefficient during confined submerged liquid jet impingement with fluo roinert ( FC 77) as t he working fluid. Webb and Ma [33 ] presented a comprehensive review of studies on jet impingement heat transfer. Ma et al [34] reported experimental measurements for local heat transfer coefficient during impingement of a circular jet perpendicular to a target plate. Both confined and fre e jet configurations were used. Garimella and Nenaydykh [35], Li et al. [36], and Fitzg erald and Garimella [37 ], all considered a confining top plate for a submerged liquid jet. Their studies used FC 77 as the working fluid at different volumetric flow rat es. Morris and Garimella [38] computationally investigated the flow fields in the orifice and the confinement regions of a normally im pinging, axis ymmetric, confined and subme rged liquid jet. Tzeng et al. [39] numerically investigated confined impinging turbulent slot jets. Eight turbulence models, including one standard and seven low Reynolds number kmodels were employed and tested to predict the heat transfer performance of multiple impinging jets. Inoue et al. [40, 41] experimentally investigated and proposed conceptual designs for the cooling of the diverte r under critical heat flux (CHF) loads for twodimensional confined planar jet on flat and concave surfaces as a function of distance from the center, flow velocity and curvature. The obtained resul ts show that the centrifugal force on the concave surface under CHF is not significant due to an existence of counter wall to suppress the splash of liquid film, which is quite different from planar jet cooling with free surface. Li and Garimella [42 ] experimentally investigated the influence of fluid thermophysical properties on the heat transfer from confined and submerged impinging jets. Generalized correlations for heat transfer were proposed based PAGE 23 9 on their results. Rahman et al. [43] numerically evaluated the conjugate heat transfer of a confined jet impingement over a stationary disk using liquid ammonia as the coolant. Ichimiya and Yamada [44] studied the heat transfer and flow characteristics of a single circular laminar impinging jet including buoyancy effect in a narrow space with a confining wall. Temperature distribution and velocity vectors in the space were obta ined numerically. Dano et al. [45 ] investigated the flow and heat transfer characteristics of confined jet array impingement with c ross flow. Digital particle image velocimetry and flow visualization were used to determine the flow char acteristics. Rahman and Mukka [46] developed a numerical model for the conjugate heat transfer during vertical impingement of a two dimensional (slot) submerged confined liquid jet using liquid ammonia as the working fluid. Robinson and Schnitzler [ 47 ] experimentally investigated the heat transfer and pressure drop characteristics of liquid jet arrays impinging on a heated surface for both confinedsubme rged and free surface flow configurations. For the submerged jet arrays a strong dependence on both jet to target and jet to jet spacing was found and correlated to adequately predict the experimental measurements. Their results revealed that submerged an d free jet configurations are not susceptible to changes in heat transfer when the nozzle is in close proximity (2 H/ dn 3) to the heated surface. Conversely, their results showed how the heat transfer deteriorated monotonically with the increment of the jet to target spacing (5 H/ dn 20) and spacing between jets Whelan and Robinson [48] experimental ly studied the cooling capabilities of a square water jet array of 45 jets under fixed jet to jet spacing and jet to target distance for six different n ozzle geometries The confined submerged jet array tests yield ed greater heat transfer coefficients when compared with their free jet array counterparts PAGE 24 10 Rahman et al. [49 ] numerically studied the heat transfer characteristics of a free liquid jet dischar ging from a slot nozzle and impinging vertically on a curved cylindrical shaped plate of finite thickness. The model included the entire fluid jet impingement region and flow spreading out over the convex plate under a uniform heat flux boundary condition. Computations were done for a series of parameters such as: jet Reynolds numbers, nozzle to target spacing ratios, inner plate radius of curvature plate thickness and for different nozzle widths using water, fluoroinert and oil as working fluids. Their r esults were presented for dimensionless solidfluid interface and maximum temperature in the solid, including local and average Nusselt numbers. Numerical simulation results were validated by comparing with experimental measurements Chang and Liou [50] presented an experimental study of heat transfer of impinging jetarray onto concave and convex dimpled surfaces with effusion. The results obtained showed the enhancement in heat transfer by each dimpled surface with and without effusion. From the above literature review it can be noticed that even though confined jet impingement heat transfer has been qui te extensively investigated, most of these are for flat surfaces O nly a few attempted to produce local heat transfer distribution of concave or convex surfaces in combination with twodimensional confined liquid jet impingement. In addition, none of the studies have attempted to explore conjugate heat transfer effect of a convex surface during twodimensional confined liquid jet impingement. Therefore, the intent of this research is to carry out a comprehensive investigation of local conjugate heat transfer with a steady flow for a two dimensional confined liquid PAGE 25 11 jet impingement over flat, concave, and convex surfaces. Comput ations using water (H2O) as the working fluid were carried out for several combinations of geometrical surfaces, a variety of jet Reynolds numbers, different solid thickness to curvature ratios four channel spacing ratios, and four radii of curvature of both concave and convex surfaces. The thermal conductivity effect was studied with the implementation of four different disk materials: copper, silicon, aluminum and Constantan. Results offer a better understanding of the fluid mechanics and heat transfer behavior of confined liquid jet on bodies with a current boundary Even though no new numerical technique has been developed, results obtained in the present investigation are entirely new. The numerical results showing the quantitative effects of differe nt parameters as well as the correlation for average Nusselt numbers will be practical guides for engineering design PAGE 26 12 C hapter 2 : Heat Transf er Analysis of Wavy Fin Assembly with Dehumidification 2.1 Physical Description of Wavy Fin Heat Exchangers The most widely used types of condensers and evaporators are shell and tube heat exchangers and finnedcoil heat exchangers ( Fig ure 2.1). Figure 2.2 shows the schema tic diagram of the evaporator. In the air conditioning system, the most important heat exchanger is the evaporator, because the useful processes of a refrigeration cycle occur in the evaporator Now days the coolant fluid on the automobile radiator is glycol (a ntifreeze ) because it has high efficiency in removing heat from the car engine. F igure 2.1 Most evaporator uses in air condition systems. PAGE 27 13 Figure 2.2 Schematic diagram of evaporator. In real life there are too many different types of fin evaporators such as square, rectangular, longitudinal, radial, and wavy as shown in Figure 2.3. Figure 2.3 Some types of fins. In general the wavy fin is more efficient because it has more area, as shown in Figure 2.4. The current work represents part of a wavy fin (Figure 2.5). Because of the axisymmetric model, we assume that the fin tip is insulated or dT/dR = 0 when R equal to RT, the results compared with uninstalled fin tip under the same conditions. PAGE 28 14 Figure 2.4 Side views of a wavy fin assembly. Figure 2.5 Side views of the physical wavy model PAGE 29 15 F urthermore the wavy fin has been converted to straight radial fin, by taking the equivalent length of the wa vy fin and using it as a real length of the straight radial fin, as shown in Figure 2.6. In addition, some calculations have been done for some types of fin. The dimensions of a real wavy fin of current work are shown in Table 2.1 and Table 2.2. Also Table 2.2 show that there are two types for surface s treatment, such as un coated surface (present model), and Hydrophilic coating. Hydrophilic coating has an affi nity to water and is usually charged or has polar side groups to their structure that will attract water. Figure 2.6 Side views of the physical street radial model. PAGE 30 16 Table 2.1 Geometr ic dimensions of sample wavy fin and tube heat exchangers No Do (mm) Dc (mm) P T (mm) P L (mm) Fp (mm ) f (mm ) N 1 9.53 9.76 25.4 19.05 1.41 0.115 2 2 9.53 9.76 25.4 19.05 1.81 0.115 2 3 9.53 9.76 25.4 19.05 2.54 0.115 2 4 9.53 9.76 25.4 19.05 2.54 0.115 4 5 9.53 9.76 25.4 19.05 2.54 0.115 6 6 9.53 10.03 25.4 19.05 1.41 0.250 2 7 9.53 10.03 25.4 19.05 1.81 0.250 2 8 9.53 10.03 25.4 19.05 2.54 0.250 2 9 9.53 10.03 25.4 19.05 2.54 0.250 4 10 9.53 10.03 25.4 19.05 2.54 0.250 6 Note: Tubes are made of copper with a wall thickness of 0.3 mm. PAGE 31 17 Table 2 .2 Geometric dimensions of sample fin and tube heat exchangers No Dc (mm) P T (mm) P L (mm) Fp (mm) f (mm) N Surface treatment Fin type 1 7.64 21 12.7 1.27 0.115 2 Un coated Slit 2 7.64 21 12.7 1.28 0.115 2 Hydrophilic coating Slit 3 6.93 17.7 13.6 1.21 0.115 1 Un coated Plain 4 6.93 17.7 13.6 1.99 0.115 1 Un coated Plain 5 7.53 21 12.7 1.23 0.115 2 Hydrophilic coating Plain 6 7.53 21 12.7 1.23 0.115 2 Un coated Plain 7 7.53 21 12.7 1.78 0.115 2 Hydrophilic coating Plain 8 7.53 21 12.7 1.78 0.115 2 Un coated Plain PAGE 32 18 2.2 Mathematical Model In the current study we consider a wavy fin assembly of uniform cross section and pitch under wet condition, as shown in Fig ure 2. 4. The water condenses at the surface as filmwise, dropwise, or mixed mode when a humid air contacts to the surface at below its dew point temperature. The differences between them depend on the surfaces. For i nstance, clean surfaces tend to promote filmwise, and treated surfaces dropwise, condensation. The created film is greatly thinner than the boundary layer in the dehumidification process, this makes the condensate thermal resistance to heat transfer flow negligible. Consider a uniform heat exchanger wavy fin attached to a plane wall, as shown in Fig ure 2.5. To complete the development of the formulation model, simplifying assumptions are made as follow s : 1. The heat flow in the fin and the temperature at any point on the fin remain constant with the time. 2. The fin material is homogenous; its thermal conductivity, the condensate film, and the wall are constant. 3. There is no contact resistance between fins in the configuration or between the fin at the base of the configuration and the prime surface. 4. The convective heat transfer coefficients between the fin and the surrounding medium are uniform and constant over the entire surface of the fin. 5. The temperature of the medium surrounding the fin is uniform. 6. The fin width is so small compared with its height that temperature gradients across the fin width may be neglected. 7. The temperatur e of the base of the fin is uniform. PAGE 33 19 8. There are no heat sources within the fin itself. 9. Heat transfer to or from the fin is proportional to the temperature excess between the fin and the surrounding medium. 10. Condensation occurs when the surrounding air dew point temperature is reached. Assume that the fin is dissipating heat to the surroundi ng environment at temperature Ts, heat is transfer ring from cold fluid T1, and the temperature distribution at any point is T(x). Because there is no heat generation in ste ady state, the energy required for heat entering and leaving the element ( must equal the heat dissipated by convection over the two fin faces, each with area (L convective dissipation is At steady st ate condition for one dimension with no heat generation, the energy balance through the wall becomes : (1) 0 dx d2 w 2 The heat transfer by conduction is equal to the difference between the heat entering and leaving the elements, according to Fourier law. (2) dx dT kA dx dT kA x x x x x x sdT d and T T(x) For The minus sign in the Fourier law means the direction of heat flow is in a direction opposed to the positive sense of the coordinate system: (3) dx dT kA qx PAGE 34 20 The total heat that is dissipated from the two faces of the fin over the element is equal to: (4) )h w (w m ]q T h[T fg s x a s (s) (x) (5) flux heat Latent q )h w (w mc fg s x a (6) flux heat Sensible ]q T h[Ts (s) (x) Now the steady state energy balance can be used to combine equations 1 and 2 so that (7) )h w (w m ]h T h[T dx dT kA dx dT kA fg s x a fg (s) (x) x x x x (8) dx d dx dT kA dx dT kA Lim2 w 2 x x x x 0 x (9) 0 q q dx x d kAs c 2 2f The ratio of sensible to total heat transfer calculated at fin temperature is R, then: 4 equation in substitute R q q q q q q Rs s c s c s (10) 0 R A k q dx x df f s 2 2f We use the manipulation to develop energy balance from simultaneous heat and mass transfer from the humid air to the condensate film. (11) 0 (x) R P B dx x df 2 i 2 2f f PAGE 35 21 The boundary conditions are the following. At x = 0 (12) ) T 2ph(T dx dT kA1 w w (13) T T T T k hpdx d T T2 1 2 w w w 2 1 (14) T T T T T T T T k ph pdx (x) d2 1 2 1 2 1 2 w w w (15) (x) 1 B pdx (x) dw i w At X = R0, (16) (z) (x) f w (17) (x) p 1 R B pdz (z) d kp pdx (x) dw b i2 f w At R= RT (18) 0 dR d f The ratio of the sensible heat flux to the total heat flux qs/(qs + qc) is given by the equation: PAGE 36 22 (19) w w h h T T h 1 q q R1 2 2 m 1 2 fg ma t s The input temperature and relative humidity were used to determine the dewpoint and the rate of condensation by using standard psy chrometric equations (ASHRA E [51 ]). The overall fin efficiency, is defined as the ratio of the actual total heat transfer rate to the maximum total heat transfer rate, (20) q q max fin In this case the fin performance is determined by a combination of heat and mass transfer. The actual total heat transfer, qfin must include both the sensible heat transfer and the latent heat transfer originated by mass transfer (condensation). The sensible heat transfer is due to convection from the air to the fin because of the temperature difference between the air and the fin and the latent heat transfer is caused by the humidity ratio difference between the air and the fin surface. The maximum heat transfer rate, qmax corresponds to an ideal fin whose surface temperature equals the temperature at the fin base under wet condi tions. PAGE 37 23 2.3 Results and Discussion This section describes the heat transfer characteristics of the mathematical model used to perform numerical simulation for conditions found in a typical air conditioner coo lin g coil under wet condition. T he integrati on of differential equations worked out by the Range Kutta method with shooting technique [ 52] corresponds to a characteristic direct expansion cooling coil used in air conditioning applications, some values are kept constant in all simulations such as, Bi1 = 1.0, Bi2 = 0.1, K =1.0, K1=0.004, P = 0.25, W = 0.5, = 2 P = 0.1 P These values were chosen using heat transfer coefficients and geometric parameters. Various values of RH, T1, and T2 are represented i n Figures 2.7 2.9 as a dimensionless temperature versus a dimensionless distance. Figure 2.7 represents the variations of dimensionless temperature with dimensionless distance for changes in the relative humidity. It could be seen that an increase in rel ative humidity decreases with dimensionless temperature, The force of water vapor diffusion increases at a larger relative humidity, and as a result so do the number of molecules of water condensing on the fin surface. A lso a higher latent heat transfer and lower temperature at the fin surface occurs. The figure also demonstrates the significant benefits of water vapor condensation during the heat transfer process when the temperature profile is compared to that for a dry c ondition (zero relative humidity). PAGE 38 24 Figure 2.7 Variation of dimensionless temperature distribution with the variation in relative humidity. Variations of dimensionless temperature with dimensionless distance for changes in the cold fluid temperature T1 can be seen in Figure 2.8. The figure shows that the fin temperature increases when T1 increases, and this le a ds to a decrease in the temperature difference between the fin and its surroundings Thus, both heat and mass transfer decrease. The condensation comes to an end when T1 is increased to a value above the air dew point temperature. It can be not ed that although the local temperature at the wall and the fin changes with T1, the change in the dimensionless temperature is insignificant. Also there i s a large over prediction of the temperature when the fin is assumed to remain dry during the heat transfer process. 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 1 1.2 1.4 1.6( T T2) / ( T1T2)Dimensionless Distance Dry RH = 50 % RH = 60 % RH = 70 % PAGE 39 25 Figure 2.8 Variation of dimensionless temperature distribution with variation in cold fluid temperature at relative humidity 50% The dimensionless temperature as a function of the dimensionless distance for the variation in the surrounding temperature T2 is shown in Figure 2.9. It was noted that an increase in the air side temperature increases the heat transfer rate in the wet fin, and also the dimensionless temperature at the wall as well as in the fin decreases with the increase in T2. Pure conduction causes a linear temperature at the wall, after which a larger slope of temperature curve is seen at the fin because of lateral convection. At constant relative humidity ai r dry bulb temperature converts to moisture content (humidity ratio). Consequently, both sensible and latent heat transfer increase when at higher 0.59 0.64 0.69 0.74 0.79 0.84 0.89 1 1.2 1.4 1.6( T T2) / ( T1T2)Dimensionless Distance Dry at T1 = 5 T1 = 5 T1 = 0 T1 = 5 PAGE 40 26 temperatures Figure 2.9 shows a plot of the dimensionless temperature for artificial dry conditions when the effects of condensation have been ignored, and also shows the discrepancy in the temperature distribution in the fin between wet and dry conditions. The latent heat transfer due to condensation is a significant portion of the total heat transfer and should not be ignored in any cooling coil design. PAGE 41 27 Figure 2.9 Variation of dimensionless temperature distribution with variation in surrounding air dry bulb temperature at relative humidity 50% 0.52 0.57 0.62 0.67 0.72 0.77 0.82 0.87 1 1.2 1.4 1.6( T T2) / ( T1T2)Dimensionless Distance Dry at T2 = 25 T2 = 25 T2 = 30 T2 = 35 PAGE 42 28 Figure 2. 10 shows the variation of dimensionless temperature distribution as a function of the dimensionless distance under two conditions, fin tip with and without insulation at constant relative humidity 50%. It was observed that by leaving the fin tip with no insulation, the area of the surface is increased, which causes better heat dissipation by increasing the fin performance. It can be seen that at insulation fin tip the heat dissipat ion is less. Figure 2.10 The present model with and without insulation in the fin tip, and at 50% RH. Co mparison of rectangular and wavy models for dry bulb temperature and 50% relative humidity RH between wavy model and the model presented by Kazeminejad [6] can be seen in Figure 2.11 A linear relationship between dry bulb temperature and 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 1 1.2 1.4 1.6( T T2) / ( T1T2)Dimensionless Distance The present model with insulation at the fin tip The present model with no insulation at the fin tip PAGE 43 29 humidity ratio was assumed in this comparison. It can be observed that there is more agreement in wavy model than in their model. Figure 2.12 shows the c ompa rison between the present model with the models of Kazeminejad [6] and Rosario and Rahman [10]. Th is comparison shows results of 1 D models for dry and 50 percent relative humidity. Constant thickness film on the fin surface was presented in wavy model. It can be noted that all models show th e same tendency of decreasing dimensionless temperature with an increase of relative humidity because of the increase of latent heat transfer due to condensation. Rosario and wavy model represent superior results than the Kazeminjad model and this demonstrate s that to achie ve excellent fin performance, one has to design the fin in a radial shape (Rosario) or wa v y shape (wavy model). Figure 2.11 Comparison of rectangular and wavy models for dry and 50% RH 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6( T T2) / ( T1T2)Dimensionless Distance Dry Rect.,(Kazeminejad [6]) RH 50% Rect.,(Kazeminejad [6]) Dry Present Model RH 50% Present Model PAGE 44 30 Figure 2.12 Comparison between present model and Kazeminejad [6] ; and Rosario and Rahman [10]. Computational results for the heat transfer of the fin assembly with and without dehumidification for various values of T1, T2, and RH are plotted in Figures 2.13 2.15. The effects of varying these conditions can be studied by plotting (AUG)dry/(AUG)wet. The ratio of heat transfer in a finned assembly to heat transfer from the bare tube surface without any fin is defined as augmentation factor. The comparison of enhancement obtained from fins under dry and wet conditions is represented by th e ratio of au gmentation factor (i n other word s, the comparison of the efficiency of a fin assembly with and without condensation at the surface ) It can be seen that the value of (AUG)dry/(AUG)wet increases with distance and reaches a constant value at dimensionless 0.18 0.28 0.38 0.48 0.58 0.68 0.78 1 1.1 1.2 1.3 1.4 1.5 1.6( T T2) / ( T1T2)Dimensionless Distance Kazeminejad [6] Present Model Rosario and Rahman [10] PAGE 45 31 di stance after 3.5. The value of the augmentation ratio is always greater than 1, demonstrating the descent of fin efficiency with condensation. The i nsignificant influence of the refrigerant temperature (T1) on the augmentation ratio is shown in Figure 2.13. The fin efficiency increases in overall heat transfer rate, although it is represented by the fin assembly redu ced with condensation. From Figure 2.14 and Fig ure 2.15, both dry bulb temperature and relative humidity increases significantly with the incr ease in the augmentation ratio; simultaneously the fin efficiency decreases with more condensation at the fin surface. Figure 2.13 (Aug)dry/(Aug)wet variation with change in T1. 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.5 2 2.5 3 3.5 4 4.5(Aug)dry/(Aug)wetDimensionless distance T1 = 5 T1 = 0 T1 = 5 PAGE 46 32 Figure 2.14 (Aug)dry/(Aug)wet variation with change in T2. 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.5 2 2.5 3 3.5 4 4.5(Aug)dry/(Aug)wetDimensionless distance T2 = 24 T2 = 27 T3 = 30 PAGE 47 33 Figure 2.15 (Aug)dry/(Aug)wet variation with change in RH The present model and the 2D mo del show the same trend as shown in Fig ure 2.16. It can be seen that the pure conduction in the tube region creates a linear dimensionless temperature, and nonlinear in the fin region because of the convection at th e fin surfaces. The RH increase creates c ondensation on the fin surfaces, as a result of which th e latent heat transfer increase causes a decrease in the dimensionless temperature. In addition, the 2 D radial geometry is a better representation of the actual cooling coil configuration in most insulation Rosario, and Rahman [ 53], because it shows a behavior closer to a real heat exchanger. The comparison of wavy model and the rectangular model under the same conditions can be seen in F igure 2.17. The result of the rectangular model shows a lower heat transfer rate because it has less area. Thus, increasing the fin area is desirable in 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.5 2 2.5 3 3.5 4 4.5(Aug)dry/(Aug)wetDimensionless distance RH = 50 RH = 57 RH = 65 % PAGE 48 34 order to obtain better fin performance, but there are some physical limitations to build ing such a fin arrangement. The results demonstrate that the fin performance in the wavy fin depends on the area of the fin, which also indicate s that the wavy fin has better performance than the rectangular one which has less area. Figure 2.16 Comparison of 1D and 2 D radial models for dry and 50% RH. 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6( T T2) / ( T1T2)Dimensionless Distance 1 D Dry 1 D RH = 50 % 2D Dry 2D RH = 50 % PAGE 49 35 Figure 2.17 Comparison of the wavy model and the converted rectangular model at dry and 50% RH. 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 1 1.1 1.2 1.3 1.4( T T2) / ( T1T2)Dimensionless Distance Rectangular Fin Wavy Fin PAGE 50 36 C hapter 3 : Conjugate Heat Transfer Analysis of a Confined Liqu id Jet Impingement on Concave and Convex Surfaces 3.1 Modeling and Simulation Figure 3.1 Two dimen s ional liqu id jet impingement on a uniform ly heated concave surface The physical model corresponds to a two dimensional confined liquid jet that impinges on a solid curved surface of circular shape, as shown in F igure 3.1. The jet discharges from the nozzle and impinges perpendicularly at the center and top of the PAGE 51 37 curved body while its bottom is subjected to a constant heat flux. The fluid is Newtonian and the flow is incompressible and symmetric about the mid plane under a steady state condition. The / z terms can be omitted as a result of this two dimensional analysis. The variation of fluid properties with local temperature is taken into account. The equations describing the conservation of mass, momentum (x and y directions respectively), and energy using a Cart esian coordinate system can be written (check Burmeister). (21) 0 y v x u (22) x y u y y x u 3 2 x u 2 x x p y u x u u f f f v v v (23) x y u x y x u 3 2 y 2 y y p y x u f f f v v v v v v y T k y x T k x y T Cp x T Cp u f f f f f f f f fv (24) y x u 3 1 y u x 2 1 y x u 22 2 2 2 f v v v The variation of thermal conductivity of solids with temperature is not significant. Therefore, the conservation of energy inside the solid can be characterized by the following equation: ) 25 ( 0 y T x T2 S 2 2 S 2 The following boundary conditions are used to complete the physical problem formulation. PAGE 52 38 (26) 0 x T : R y H 0, x AtS O (27) 0 x T 0, x 0, u : 0 y R 0, xf O v j f j(28) T T V 0, u : 2 w x 0 0 y At v (29) 0 x T 0, u : 0 y R 2 w x Atf i v 0 T 0, u : ) 2 w (x curvature, inner At f s v i R r r T k r T k T T 0, u : ) R (r curvature outer At f f S S f S 0 v ) 30 ( y T k : R x 0 H y AtS S o q (31) p p : R x R 0, y Atatm O i (32) 0 x T 0 u : 0 y R x AtS O v H The local heat transfer coefficients can be defined as: (33) T T 1 hj int The average heat transfer coefficient can be calculated by integrating the local distributing results in the following equation (34) d T T h T T 1 hmax0 j int j int max avg Here, intT is the average temperature at the solid liquid interface. The average temperature is calculated by taking the areaweighted average of the local interface PAGE 53 39 temperature. The local and average Nusselt numbers are calculated according to the following expressions: ) 35 ( k w h Nuf ) 36 ( k w h Nuf avg avg The governing equations (1 5) along with the boundary conditions (6 14) are solved using the Galerkin finite element method as demonstrated by Fletc her [54 ]. Four node quadrilateral elements are used. In each element, the velocity, pressure, and temperature fields are approximated which leads to a set of equations which define the continuum. The number of elements required f or accurate results is determined from a grid independence study. A structured grid is used in which the size of the elements near the solid fluid interface is made smaller to adequately capture large variations in velocity and temperature in that region. The solution of the resulting nonlinear differential equations is carried out using the NewtonRaphson method. Due to the non linear nature of the governing transport equat ions, an iterative procedure is used to arrive at the solution for the velocity and temperature fields. The solution is considered converged when the field value does not change from one iteration to the next and the sum of the residuals for all the depend ent variables is less than a predefined tolerance value; in this case, 106. The values of Reynolds number is limited to a maximum of 2000 to stay within the laminar region. The nozzle opening and the solid plate have a length of 3 and 30 mm respectively. The heat flux (q) is kept constant at a value of 125 kW/m2. The incoming PAGE 54 40 fluid jet temperature (Tj) is 310 K for water. The base thickness of the solid plate (b) is varied over the following values: 10, 20, 30, 40, 50 mm. The channel spacing height or gap is set to the following values: 1, 2, 3, 4, 5 m m The radius of curvature (RO) is extended from 30 to 34 mm. The range of Reynolds number is varied from 750 to 2000. Al l runs used in the paper check out to be laminar. The simulation is carried out for a number of disk materials: aluminum, Constantan, copper, and silicon. The properties of solid materia ls are obta ined from zisik [55 ]. Fluid properties for H2O are obtained from Bejan [56] The properties of the above fluids are correlated accord ing t o the following equations: water between 300 K < T < 411 K; Cpf = 9.5x103.T2 5.9299.T + 5098.1; kf = 7.0x106.T2 + 5.8x103.T 0.4765; f = 2.7x103.T2 + 1.3104.T + 848.07; and ln( f) = 3.27017 0.0131.T. PAGE 55 41 3.2 Results and Discussion This section describes the heat transfer characteristics of a confined liquid jet impingement under flat, concave, and convex surfaces. The velocity ve ctor distribution remains uniform at the potential core region of the confined liquid jet through the curvature as shown in Figure 3.2. The direction of motion of the fluid particles shifts by more than 90 in a concave surface, 90 in the flat surface, and less than 90 in the convex surface. Figure 3. 2 Ve locity vector distribution for jet imping e ment on a curved copper plate. VELOCITY VECTOR UNIT SCALE (cm/s) 113.1 84.8 56.5 28.3 14.1 6.21 cm 9.315 cm 6.21 cm VELOCITY VECTOR UNIT SCALE (cm/s) 113.1 84.8 56.5 28.3 14.1 6.21 cm 9.315 cm 6.21 cm PAGE 56 42 Thereafter, the fluid strikes the solid surface at which point there is a rapid deceleration while the flow chan ges direction along the surface After this, there is a brief ac celeration starting the development of boundary layer. It can be noted that the boundary layer thickness increases along the radius of curvature, and the frictional resistance from the wall is eventually transmitted to the fluid flow. The fluid between th e boundary layer zone and confined top plate has much smaller flow velocity compared to the inlet velocity. This is due to frictional resistance from the solid body, as well as the confined plate. Figure 3. 3 Solid fluid interface temperature for different number of elements in x and y directions ( Re = 1,000, b = 30, w = 0.6 cm) The solidfluid dimensionless interface temperatures for different number of grids are plotted in F igure 3 .3 Several grids are used to determine the number of elements needed for accurate numerical solution. It is observed that the numerical solution becomes grid independent when the grids reach a number of divisions equal to 12x130 in 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2 4 6 8 10Dimensionless Interface Temperature,intSolid Fluid Interface Distance, S(cm) 4x66 6x80 6x150 8x99 9x150 10x195 12x130 20x210 PAGE 57 43 (y) and (x) directions respecti vely. Numerical results for a 12x130 grid gave almost i dentical results compared to 10x195 and 9x150 grids for an impingement height (hn) equal to 30 mm. Therefore, the chosen grid is 12x130, which carries an average margin error of 0.163%; all further computations are carried out using this grid distribution. The size of the elements varies with denser distribution at the solidfluid interface and at the nozzle axis. Figure 3.4a and F igure 3.4b show the variation of solidfluid dimensionless interface temperature plots and local Nusselt number distributions a t different Reynolds number s for concave and convex surfaces respectively, with water as a cooling fluid and copper as the solid body material. The plots reveal that dimensionless interface temperature decreases with jet velocity (or Reynolds number) for e ither type of plate configu ration. At any Reynolds number, the dimensionless interface temperature has a low value at the stagnation point and increases radially along the radius of curvature reaching the highest value at the solid fluid interf ace distance (S) ( approximately equal to 2.52 cm ) and decreases to its lowest value at the end of the concave curvature, as shown on F igure 3.4a. A new behavior occurs along the upright concave surface, causing the dimensionless temperature to drop. This is due to an energy balance, where more of the heat dissipates at the interface along the jet impingement region that is closer to the base of the plate under a uniform heat flux boundary condition that gradually move s far away at constant flow rate conditions. At this condition t he thickness of the thermal boundary layer de creases along th e radius of curvature causing the i nterface temperature to drop along the radial distance T his allows the heat to dissipate faster and results in a lower interface temper ature at the end of the concave plate. PAGE 58 44 (a) (b) Figure 3.4 Dimensionless interface temperature and Local Nusselt number distribution for (a) concave and (b) convex copper plate at different Reynolds number s and water as the cooling f1uid. 12 24 36 48 60 72 84 96 108 120 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 Solid Fluid Interface Distance, S(cm) Re =750 Re =1000 Re =1250 Re =1500 Re =1750 Re =2000 Nu, Re=750 Nu,Re=1000 Nu, Re=1250 Nu, Re=1500 Nu, Re=1750 Nu, Re=2000 Local Nusselt Number, Nu Dimensionless Interface Temperature, int 12 24 36 48 60 72 84 96 108 120 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 Solid Fluid Interface Distance, S(cm) Re =750 Re =1000 Re =1250 Re =1500 Re =1750 Re =2000 Nu, Re=750 Nu, Re=1000 Nu, Re=1250 Nu, Re=1500 Nu, Re=1750 Nu, Re=2000 Local Nusselt Number, Nu Dimensionless Interface Temperature, int PAGE 59 45 Conversely, the dimensionless interface temperature for the convex plate has the lowest value at the stagnation point (underneath the center of the axial opening) and increases radially downstream reaching the highest value at the end of th e curvature, as shown in Figure 3.4b. The thickness of the thermal boundary layer increases along the radius of curvature of the convex plate and causes the interface temperature to increase due to the proximity of the solidfluid interface to the heat flux boundary condition. Local Nusselt number distributions of Figure 3.4a are half bell shaped with a peak at the stagnation point and gradually increase along the concave surface, reaching the highest value at the end of the radius of curvature. Contrar il y, all local Nus selt number distributions of Figure 3.4b show a half bell profile with a peak at the stagnation point and a decrease along the radius of curvatur e of the convex plate. Figures 3.4a and 3.4b confirm how an increasing R eynolds number contributes to more effe ctive cooling. Si milar profiles shown in Figure 3.4b have been documente d by Ma et al. [34], and Garimella and Nenaydykh [35]. Figures 3.5a and 3.5b present the average Nusselt number as a function of Reynolds number and different radius of curvature. It c an be seen that the average Nusselt number increases according to the Reynolds number. As the flow rate (or Reynolds number) increases, the magnitude of fluid velocity near the solidfluid interface that controls the convective heat transfer rate increases Furthermore, at a particular Reynolds number, the Nusselt number decreases with the increment of the radius of curvature. In figure 3.5b we can see that at radius 7.01 cm the average Nussselt number is highest, this because the concave is more closer to the heat flux. PAGE 60 46 (a) (b) Figure 3. 5 A verage Nusselt number at different Reynolds number s for (a) concave (b) convex copper plate with water as the cooling fluid (R= 6.21, 6.61, 7.01, and ,cm) 15 17 19 21 23 25 27 29 31 33 35 500 1000 1500 2000 2500Average Nusselt NumberReynolds number R = 6.21 cm R = 6.61 cm R = 7.01 cm Angle = Zero R = 6.21 cm R= 6.61 cm R= 7.01 cm R = 15 25 35 45 55 65 75 500 1000 1500 2000 2500Average Nusselt NumberReynolds number R = 6.21 cm R = 6.61 cm R = 7.81 R = Infinity R = 6.21 cm R= 6.61 cm R= 7.01 cm R = PAGE 61 47 In addition, it can be seen that the a verage Nusselt number plots get closer to each other as the radius of curvature decreases. This behavior confirms the positive influence of the radius of curvature ( ) on the average Nusselt number down to =62.1, which corresponds to an outer radius of curvature of 6.2 1 cm. The radius of curvature effects on the dimensionless interface temperature and loc al Nusselt number are shown in Figure 3.6a and F igure 3.6b for concave and F igures 3.7a and 3.7b for convex. The dimensionless s olid fluid interface distance increases for the concave from the impingement region all the way to the end at the infinite radius, and increases to the peak point at the highest solid thickness region and drops down to the lowest at the shortest solid thickness for other radiuses as s hown in F igure 3.6a. We obs erve in F igure 3.7a better results for convex during the increase in temperature from the impingement region all the way to the end at all radiuses. The higher outflow temperature occurs when the temperature is lower at the stagn ation region. This is fairly estimated since the total heat transferred to the curvature as well as the fluid flow rates are the same for all cases. For the concave as shown in Figure 3.6b, the local Nusselt number decreases with the solidfluid interfac e distance for a rate of radius of curvature ( ) from 31.05 at Reynolds number of 1000 at maximum of thickness and start s increasing to highest at the minimum of thickness. PAGE 62 48 (a) (b) Figure 3.6 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for a concave copper wafer at different radius, and water as the cooling f1uid (R= 6.21, 6.61, 7.01, and cm) 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 2 4 6 8 10Dimensionless Interface Temperature, intSolidFluid Interface Distance, S(cm) R = 6.21 cm R = 6.61 cm R = 7.01 cm R = 20 25 30 35 40 45 0 2 4 6 8 10Local Nusselt Nummber,NuSolidFluid Interface Distance, S(cm) R = 6.21 cm R = 6.61 cm R = 7.01 cm R = Infinity = 31 = 33 = 35 = = 31 = 33 = 35 = PAGE 63 49 (a) (b) Figure 3.7 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for a convex copper wafer at different r adius, and water as the cooling f1uid (R= 6.21, 6.61, 7.01, and cm) 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0 2 4 6 8Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) R = 6.21 cm R = 6.61 cm R = 7.01 cm R = Infinity R = 6.21 cm R= 6.61 cm R= 7.01 cm R = 20 22 24 26 28 30 32 0 2 4 6 8Local Nusselt Number, NuSolid Fluid Interface Distance, S(cm) angle = zero R = 7.01 cm R 6.21 cm R = 6.61 cm R = 6.21 cm R= 6.61 cm R= 7.01 cm R = PAGE 64 50 Figure 3.8b illustrates superior consequences for convex compare d with concave throughout the decrease of Nusselt number all the way to the end, witho ut changing due to the difference of thicknesses. The local fluid velocity adjacent to the heated materia l surface creates an enhancement of Nusselt number due to the confined impingement jet. Co pper has been used as the solid material and water as the cooling fluid for a Reynolds number of 1000 and solid thickness to curvature ratio of 0.161 0.5. The dif ference of solid thickness to curvature spacing ratios ( ) from 0.161 0.5 are modeled for water as the coolant and co pper as the solid material. The effects of solid thickness to the spacing of curvature on the local Nusselt number and dimensionless inte rface temperature at a Reynolds numbe r of 1000 are shown in F igures 3.8a and 3.8b. It may be noted that the solid thickness insignificantly affects the local Nusselt number distribution particularly at the end; however there is a minor change at the stagna tion region. PAGE 65 51 (a) ( b) Figure 3. 8 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different material thickness (H = 1, 1.5, 2, 2.5, and 3 cm) 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0 2 4 6 8 10Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) H = 1 cm H = 1.5 cm H = 2 cm H = 2.5 cm H = 3 cm 22.5 24.5 26.5 28.5 30.5 32.5 34.5 0 2 4 6 8 10Local Nusselt Number, NuSolid Fluid Interface Distance, S(cm) H = 1 cm H = 1.5 cm H = 2 cm H = 2.5 cm H = 3 cm PAGE 66 52 The solidfluid dimensionless interface temperature and local Nusselt number distributions for five different spacing of curvature for water as the cooling fluid and Reynol ds number of 1000 are shown in F igures 3.9 (a and b) and 3. 10 (a and b) respectivel y. Due to the higher jet momentum at impingement at the end of the nozzle the temperature at the solid fluid interface decreases causing higher velocity of fluid particles adjacent to the plate enhances the heat transfer. In F ig ures 3. 9 and 3.10, a higher Nusselt number is seen all along the arc length at all radii of different spacing and also the Nusselt number increases by increasing the spacing o f curvature ( from 0.1 0.5 cm ) Also we have seen that the impingement height affects the Nuss elt number more at the stagnation region and the early part of the boundary layer region. For larger spacing (0. 5 cm ) the values get closer for al l impingement heights. Hence, it can be concluded that the jet momentum more strongly affects the areas subje cted to direct impingement. Because of the fast trav eling of heat at less material, i t can be noted t hat the Nusselt number increases at all radii of different spacing. PAGE 67 53 (a) (b) Figure 3.9 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different spacing of concave curvature (D = 0.1, 0.2, 0.3, 0.4, and 0.5 cm) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 2 4 6 8 10 12Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) D = 0.1 cm D = 0.2 cm D = 0.3 cm D = 0.4 cm D = 0.5 cm 10 15 20 25 30 35 40 45 50 55 60 0 2 4 6 8 10 12Local Nusselt Number,NuSolid Fluid Interface Distance, S(cm) D = 0.1 cm D = 0.2 cm D = 0.3 cm D = 0.4 cm D = 0.5 cm PAGE 68 54 (a) (b) Figure 3.10 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different spacing of convex curvature (D = 0.1, 0.2, 0.3, 0.4, and 0.5 cm) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 2 4 6 8Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) D = 0.1 cm D = 0.2 cm D = 0.3 cm D = 0.4 cm D = 0.5 cm 30 40 50 60 70 80 0 1 2 3 4 5 6 7Local Nusselt Number, NuSolid Fluid Interface Distance, S(cm) D = 0.1 cm D = 0.2 cm D = 0.3 cm D = 0.4 cm D = 0.5 cm PAGE 69 55 Figures 3.11a and 3.11b show the dimensionless solidfluid interface temperature and local Nusselt number distribution plots respectively as a function of the dimensionless radial distance measured from mid plane axis for different solid materials with water as the working fluid. The numerical simulation is carried for a set of mater ials : aluminum, copper, C onstantan and silicon, which all have different thermo physical properties. The temperature distribution plots reveal how the thermal conductivity of the solids affects the heat flux distribution that controls the local interface temperature. It may be noted that C onstantan has the lowest temperature at the impingement axis and the highest at the inner radial distance of the concave plate. This large interface temperature variation is due to its lower thermal conductivity. As the thermal conductivity increases, the thermal resistance within the solid becomes lower and the interface temperature becomes more uniform as seen in the plots corresponding to copper and silicon. The cross over of the curve s of the four materials occurs due to a constant fluid flow and heat flux rate that provide s a constant thermal energy transfer for all circumstances. Narrow and elevated bell shape pattern for local Nusselt number dis tributions are seen in F igure 3.11b for all solid materials with low thermal conductivity. Conversely high thermal conductivity materials such as aluminum and copper portray a more uniform local Nusselt number distribution in general. PAGE 70 56 (a) (b) Figure 3.11 Solid fluid interface distance and (a) dimensionless interface temperature distribution (b) Local Nusselt number distribution for different materials (c opper, silicon, a luminum, and Constantan) 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0 2 4 6 8 10Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) Cupper, Cu Silicon, Si Aluminum, Al Constantan, CuNi 15 20 25 30 35 40 45 50 55 0 2 4 6 8 10Local Nusselt Number,NuSolid Fluid Interface Distance, S(cm) Cupper, Cu Silicon, Si Aluminum Constantan, CuNi PAGE 71 57 Figure 3.12a and F igure 3.12b show the variation of solidfluid dimensionless interface temperature plots and local Nusselt number distributions at different Reynolds number s for concave and convex surfaces respectively, with water as a cooling fluid and s ilicon as a material. It can be seen th at there is no significant change in the result s between copper and silicon. The s ame plots reveal that dimensionless interface temperature decreases with jet velocity (or Reynolds number) for either type of plate configuration. Also, at any Re ynolds number, the dimensionless interface temperature has a low value at the stagnation point and increases radially along the radius of curvature reaching the highest value at the solid fluid interface distance, (S) ( approximately equal to 3.00 cm ) and decrease s to its lowest value at the end of the concave curvature, as shown on F igure 3.12a. T he difference is 0.48 cm. T his behavior is due to the development of a thermal boundary layer as the fluid moves downstream from the center of the concave curvature, and the difference between the thermal conductivity of the material. The thickness of the thermal boundary layer increases along the radius of curvature and causes the interface temperature to increase; subsequently the area of curvature diminis hes along the radial distance, allowing the heat to dissipate faster, resulting in a lower interface temperature at the end of the concave plate. This also makes a difference in the local Nusselt number with in a range of about 48. On the other hand, the dimensionless interface temperature for the convex plate has the lowest value at the stagnation point (underneath the center of the axial opening) and increases radially downstream reaching the highest value at the end of the curvature as shown in F igur e 3.12b. The thickness of the thermal boundary layer increases along the radius of curvature of the convex plate and causes the interface temperature to increase. This is PAGE 72 58 caused by the proximity of the solid fluid interface due to the heat flux boundary c ondition. Local Nusselt n umber distributions of F igure 3. 12a are half bell shaped with a peak at the stagnation point and gradually increase along the concave surface, reaching the highest value at the end of the radius of curvature. Contrar il y, all local Nusselt numbe r distributions of F igure 3.12b show a half bell profile with a peak at the stagnation point and decrease along the radius of curvature of the convex plate. PAGE 73 59 (a) (b) Figure 3.12 Dimensionless interface temperature and Local Nusselt number distribution for (a) concave and (b) convex s ilicon plate s at different Reynolds number s and water as the cooling f1uid. 12 24 36 48 60 72 84 96 108 120 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 2 4 6 8 10Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) Re =750 Re =1000 Re =1250 Re =1500 Re =1750 Re =2000 Nu, Re=750 Nu, Re=1000 Nu, Re=1250 Nu, Re=1500 Nu, Re=1750 Nu, Re=2000 Local Nusselt Number Nu 12 24 36 48 60 72 84 96 108 120 0 0.02 0.04 0.06 0.08 0.1 0.12 0 2 4 6 8 10Dimensionless Interface Temperature, intSolid Fluid Interface Distance, S(cm) Re =750 Re =1000 Re =1250 Re =1500 Re =1750 Re =2000 Nu, Re=750 Nu, Re=1000 Nu, Re=1250 Nu, Re=1500 Nu, Re=1750 Nu, Re=2000 Local Nusselt Number, Nu PAGE 74 60 P apers used for the validation of this numerical study included analytical works by Inoue et al.[40] and Inoue et al.[41] using fluids with Reynolds number s between 500 200. The fluids were tested for heat removal under confined liquid jet impingement on a heated flat surface maintained at uniform heat flux. The graphical representation of actual numerical average Nusselt nu mber results at the stagnation point at different Reynol ds number s are shown in F igure 3.13. The local Nusselt number under Reynolds numbers of 750, 1,000, 1,250, 1,500, 1750, and 2000 correlates with an average difference margin of 17.95%, 12.1%, 11.11%, 10.35%, 12.7%, and 12.12% respectively The results shown in F igure 3.13 were on average within 34.29% of Rahman et al. [49] within 35% of A. Inoue et al. [ 40] and within 33.33% for the current work. Considering the in herent discretization and round off e rrors, this comparison of Nusselt number s at the stagnation point is quite satisfactory. PAGE 75 61 Figure 3.13 Stagnation Nusselt number com parison of Rahman et al. [49 ], Inoue et al. [ 40], with actual numerical results under different Reynolds numbers (w = 4 mm, d =2 mm) 19 21 23 25 27 29 31 33 35 0 500 1000 1500 2000 2500Average Nusselt NumberReynold number Rahman et al. [49] Present model Inoue et al. [40] PAGE 76 62 C hapter 4: C onclusion s The analytical model for a onedimensional wavy fin assembly under fully wet conditions has been developed. The model was considered with and without insulation at the fin tip under the same conditions. The same model has been converted to straight radial (rectangular) fin, and the results revealed that at the insulation fin ti p the heat dissipation is less. We also found that under the same operating conditions ; the radial wavy fin provides better heat transfer performance than the radial rectangular one. The cooling and dehumidification fin assembly heat transfer performance has been carried out when synchronous mass and heat transfer take place. The results show that generally the fin efficiency depend s on the condit ion of the surface and the area of the fin under wet condition. The heat transfer characteristics have been carried out at variations of T1, T2, and RH The latent heat transfer under wet condition during the condensation process enhance s th e heat transfer rate to a fin assembly when dehumidification occurs, at a rate which is always higher than the dry fin assembly. Under fully wet conditions the dimensionless temperature, decreases with temperature and relative humidity of the surroundi ng air, thus the fin efficiency changes rapidly with air relative to humidity. The study of the effects of differences in cold fluid temperature (T1), air side temperature (T2), and relative hu midity (RH) has led to a better understand ing of heat and mass transfer occurring in the air conditioning dehumidification coils. The results show that at any increase in the air side temperature (T2) while the cold fluid temperature and relative PAGE 77 63 humidity remain constant, both sensible and latent he at transfer increases at the coil. The heat and mass transfer decrease by increasing the fin temperature when the cold fluid temperature (T1) increases, and the air side temperature (T2) and the relative humidity (RH) remain constant. Due to a larger conde nsation rate at the fin surface, the dimensionless temperature decreases when the relative humidity increases. At all results, the heat transfer rate of the fin assembly is higher than that of a dry fin assembly when dehumidification occurs. The variations of cold fluid temperature (T1) enhance the augmentation factor of the wet fin assembly compared to the dry surface condition. The increase in the amount of dehumidification makes a reduction in the wet augmentation factor. The increment in the area of the fin surface, air side temperature (T2), and the relative humidity (RH), illustrate the increase in the ratio of the dry to wet augmentation factor. The findings of the current work demonstrate that the overall fin efficiency is dependent on the relative humidity of t he surrounding air and the area of the fin. The eff ici ency depends on the fin surface area; The increase in surface area causes better heat dissipation by increasing the fin performance However, even though an increase in fin surface area is desirable in order to obtain better fin performance, there are some physical limitations involved in building such a fin arrangement. Covering dry, partially wet, and fully wet conditions give s us a complete understanding of heat transfer phenomenon for an efficient design of dehumidification apparatus The solidfluid dimensionless interface temperature and local and average Nusselt number for concave, convex, and flat surface s show a strong dependence on Reynolds number, curvature spacing, length of radius, impingement height, and solid material properties. The increment of Reynolds number increases the local heat transfer PAGE 78 64 coefficient distribution values over the entire solidfluid interface for all different material s The result s showed that decreasing t he nozzle width increases the local Nusselt number at the core region. Decreasing the channel spacing, plate thickness or plate inner radius of curvature all enhanced the local Nusselt number. It can be seen that implementation of confined liquid jet impingement over a convex surface is more effective compared to flat or concave surface cooling methods The ongoing contrivance harvest s low cost and accurate prediction of processes which involve jet impingement cooling. This approach is useful for the design of relevant cool ing applications which enhance the heat transfer removal encountered on high heat flux of concave and convex surfaces. Numerical simulation res ults are validated by comparison with the experimental measurements of flat and concave surfaces. PAGE 79 65 R efere nces [1] Lunardini JV, Aziz A (1995) Effect of Condensation on P erf ormance and Design of E xtended S urfaces. CRREL Report 9520. [2] Kay s WM, London AL (1964) Compact Heat E xchangers McGraw Hill, New York. [3] Wang C, Hsieh Y, a nd Lin Y (1997) Performance of Plate Finned Tube Heat Exchangers Under D ehumidifying C onditions Journal of Heat Transfer Vol. 119, pp. 109117. [4] Leu J, Che n S, and Yuhjang J (2004) Heat Transfer and F luid Flow in Rectangular Fin and Elli ptic T ube H eat E xchangers U nder D ry and Dehumidifying C onditions. Journal of Enhanced Heat Transfer Vol. 11, No. 1, pp. 4360. [5 ] Webb RL (1994) Principles of Enhanced Heat T ransfer John Wiley & Sons, New York. [6] Ka zeminejad H (1995) Analysis of One Dimensional Fin A ssem bly H eat Transfer with D ehumidification International Journal of Heat and Mass Transfer Vol. 38, No. 3, pp. 455462. [7] Salah El Din MM (1998) Performance Analysis of Partially Wet Fin Assembly Applied Thermal Engineering Vol. 18, No. 5, pp. 337349. [8] Liang SY, Wong, TN, and Nathan G K (2000) Comparison of O ne Dimensional and TwoDimensional Models for Wet Surface E fficiency of a Plate Fin Tube Heat E xchanger Applied Thermal Engineering Vol. 20, pp. 941962. [9] Rosar io L, Rahman MM (1998) Overall E fficiency of a Radial Fin A ssem bly Under Dehumidifying C onditions Journal of Energy Resources Technology Vol. 120, No. 4, pp. 299304. [10] Rosario L Rahman MM (1999) Analysis of Heat Transfer in a Partially Wet R adial Fi n Assembly During D ehumidification. International Journal of Heat and Fluid Flow Vol. 20, pp. 642 648. PAGE 80 66 [1 1] Threlkeld JL (1970) Thermal Environmental E ngineering. PrenticeHall, New York. [12] ARI Standard 41081 (1972) Forced C irculation A ir Cooling and A ir H eating C oils Air Conditioning and Refrigeration Institute. [13] McQuiston FC (1975) Fin Efficiency with C ombined H eat and Mass T ransfer ASHRAE Transactions Vol. 81, No. 1, pp. 350355. [14] Coney JER, Sheppar d CGW, and El Shafei EAM (1989) Fin P erformance with Condensation from Humid A ir: a Numerical I nvestigation. International Journal of Heat and Fluid Flow Vol. 10, No. 3, pp. 224231. [15] Srinivasan V, Shah RK (1997) Fin Efficiency of Extended Surfaces in Two Phase F low International Journal of Heat and Fluid Flow Vol. 18, No. 4, pp. 419429. [16] Elmahdy AH, Biggs RC (1983) Efficiency of Extended S urfaces with Simultaneous Heat and Mass T ransfer ASHRAE Transactions Vol. 89, No. 1A pp. 135143. [17] McQuiston FC, Parker JD (1994) Heating, Ventilating, and Air C onditioning, John Wiley & Sons, New York. [18] Schmidt TE (1949) Heat Transfer Calculations for Extended Surfaces. Refrigerating Engineering pp. 351357. [19] Hong TK, Webb RL (1996) Calculation of Fin Efficiency for Wet and Dry F ins. Inte rnational Journal of HVAC and Research, Vol. 2, No. 1, pp. 2741. [20] Lin Y, Hsu K, Chang Y, and Wang C (2001) Performance of R ectangular F in in W et Conditions: V isualization and Wet F in E fficiency Journal of Heat Transfer Vol. 123, No. 5, pp. 827836. [21] Heggs P J, Ooi TH (2004) Design Charts for Radial R ectangular F ins in T erms of P erformance R atio and Maximum E ffectiveness Applied Thermal Engineering Vol. 24, pp. 13411351. [22] Lin CN, Jang JY (2002) A T wo D imensional Fin E fficiency A nalysis of Combined Heat and Mass T ransfer in E lliptic F ins International Journal of Heat and Mass Transfer Vol. 45 pp. 38393847. [23] Martin, H (1977) Heat and Mass Transfer Between Impinging Gas Jets and Solid Surfaces, Adv. Heat Mass Transfer Vol.13, pp.160. [24] Viskanta, R, (1993) Heat T ransfer to Impinging Isothermal Gas and Flames Jets Thermal and Fluid Science, Vol. 2, pp. 111134. PAGE 81 67 [25] Hong, S K, Lee, DH, and Cho, HH, (2008) Heat/Mass Transfer Measurement on Concave Surface i n Rotating Jet Impingement, Journal of Mechanical Science and Technology, Vol. 22, No. 10, pp.19521958. [ 26] Glauert, M.B. The Wall Jet, Journal of Fluid Mechanics, Vol.1, No.6, 1956, pp. 625643. [27] McMurray, D.C., Myers, P.S., and Uyehara, O.A., Influence of Impinging Jet Variables on Local Heat Transfer Coefficients Along a Flat Surface with Constant Heat Flux, Proc. of the 3rd Inter. Heat Transfer Conference Chicago, IL, Vol. 2, 1966, pp. 292299. [28] Metzger, D.E., Cummings, K.N., and Rub y, W.A., Effects of Prandtl Number on Heat Transfer Characteristics of Impinging Liquid Jets, Proc. of the 5th Inter. Heat Transfer Conference Tokyo, Vol. 2, 1974, pp. 2024. [ 29] Thomas, S., Faghri, A., and Hankey, W.L., Experimental Analysis and Fl ow Visualization of a Thin Liquid Film on a Stationary and Rotating Disk, Journal of Fluids Engineering, Vol. 113, No.1, 1991, pp.7380. [ 30] Faghri, A., Thomas, S., and Rahman, M.M., Conjugate Heat Transfer from a Heated Disk to a Thin Liquid Film form ed by a Controlled Impinging Jet, Journal of Heat Transfer, Vol.115, No.1, 1993, pp. 116123. [31] Hung, Y.H., and Lin, Z.H., Effect of Confinement Plate on Heat Transfer Characteristics of a Circular Jet Impingement , Proc. of the ASME Fundamentals of Heat Transfer in Forced Convection, HTD Vol. 285, 1994, pp.101109. [32] Garimella, S.V., and Rice, R., Confined and Submerged Liquid Jet Impingement Heat Transfer, Journal of Heat Transfer, Vol. 117, No. 4, 1995, pp. 871877. [33] Webb, B.W., and Ma, C.F., Single phase Liquid Jet Impingement Heat Transfer, Advances in Heat Transfer 26(1), 1995, pp. 105117. [34] Ma, C.F., Zheng, Q., Lee, S.C., and Gomi, T., Impingement Heat Transfer and Recovery Effect with Submerged Jets of Large Prandtl Number Liquid 2. Initially Laminar Confined Slot Jets, International Journal of Heat and Mass Transfer, Vol. 40, No.6, 1997, pp. 14911500. [35] Garimella, S.V., and Nenaydykh, B., Nozzle Geometry Effects in Liquid Jet Impingement Heat Transfer, Inter national Journal of Heat and Mass Transfer, Vol. 39, No. 14, 1996, pp. 29152923. PAGE 82 68 [36] Li, D.Y., Guo, Z.Y., and Ma, C.F., Relationship Between the Recovery Factor and the Viscous Dissipation in a Confined, Impinging, Circular Jet of High Prandtl Number Liquid, International Journal of Heat and Fluid Flow, Vol. 18, No.6, 1997, pp. 585590. [37] Fitzgerald, J.A., and Garimella, S.V., A Study of the Flow Field of a Confined and Submerged Impinging Jet, International Journal of Heat and Mass Transfer, Vol. 41, No. 89, 1998, pp. 10251034. [38] Morris G.K., and Garimella, S.V., Orifice and Impingement Flow Fields in Confined Jet Impingement, Journal of Electronic Packaging, Vol. 120, No. 1, 1998, pp. 6872. [39] Tzeng, P.Y. Soong, C.Y. and Hs ieh, C.D. Numerical Investigation of Heat Transfer Under Confined Impinging Turbulent Slot Jets, Numerical Heat Transfer, Part A, Vol.35, No.8, pp. 903924, 1999. [40] Inoue, A., Ui, A., Yamaz aki, Y., Matsusita, H., and Lee S.R., Studies on a Cooling of High Heat Flux Surface in Fusion Reactor by Impinging Planar Jet Flow, Fusion Engineering and Design Vol. 5152, 2000, pp. 781787. [ 41] Inoue, A., Ui A., Yamazaki, and Lee, S.R., Studies on Cooling by Twodimensional Confined Jet Flow of High Heat Flux Surface in Fusion Reactor, Nuclear Engineering and Design Vol. 200, 2000, pp. 317329. [4 2] Li, C.Y., and Garimella, S.V., Prandtl Number Effects and Generalized Correlations for Confined and Submerged Jet Impingement, Inter. Journal of Heat and Mass Transfer, Vol. 44, No.18, 2001, pp. 34713480. [43] Rahman, M.M., Dontaraju, P., and Ponnappan, R., Confined Jet Impingement Thermal Management usi ng Liquid Ammonia as the Working Fluid, Proc. of the ASME Inter. Mechanical Engineering Congress and Expo., New Orleans, Louisiana, 2002, pp. 110. [44] Ichimiya, K., and Yamada, Y., Three Dimensional Heat Transfer of a Confined Circular Impinging Jet w ith Buoyancy Effects, Journal of Heat Transfer, Vol. 125, No.2, 2003, pp. 250256. [45] Dano, B., Liburdy, J.A., and Kanokjaruvijit, K., Flow Characteristics and Heat Transfer Performances of a Semi confined Impinging Array of Jets: Effect of Nozzle Ge ometry, Inter. Journal of Heat and Mass Transfer, Vol. 48, No. 34, 2005, pp.691701. [46] Rahman, M.M., and Mukka, S. K., Confined Liquid Jet Impingement on a Plate with Discrete Heating Elements, Proc. of the ASME Summer Heat Transfer Conference, Vol.4, 2005, pp. 637647. PAGE 83 69 [47] Robinson, A.J., and Schnitzler, E. An Experimental Investigation of Free and Submerged Miniature Liquid Jet Array Impingement Heat T ransfer, Experimental Thermal and Fluid Science, Vol. 32, No.1, 2007, pp.113. [48] Whelan, B.P., and Robinson A.J., Nozzle Geometry Effects in Liquid Jet Array Impingement, Applied Thermal Engineering, Vol. 29, No. 1112, 2009, pp. 221122. [49] Rahman, M.M., Hernandez, C., and Lallave, J.C., Free Liquid Jet Impingement From a Slot Nozzle to a Curved Plate, Numerical Heat Transfer, Part A, Vol. 57, No. 11, 2010, pp.799821. [50] Chang, S., and Liou, H., 2009 Heat Transfer of Impinging Jet array onto Concave and Convex dimpled Surfaces with Effusion International Journal of Heat and Mass Transfer, Vol.52, 2009, pp.44844499. [ 51] Wang C, Lee S, Sheu J W, and Chang J Y (2002) A comparison of the airside performance of the fin andtube heat exchangers in wet conditions; with and without hydrophilic coating. Applied Thermal Enginee ring, v 22, n 3, p 26778. [ 52] Sharma JN (2004) Numerical methods for engineers and scientists Alpha Science International, Ltd. Pangbourne, UK. 231237. [53] L. Rosario and M.M. Rahman, A Two Dimensional Numerical Study of Heat Transfer in a Finned Tube Assembly during Axisymmetric Dehumidification, ASME Journal of Energy Resources Technology Vol.121, No.4, pp. 247253, 1999. [5 4] Fletcher, C.A.J., Computational Galerkin Methods, Springer Verlag, New York, 1984, pp. 27 and 205. [5 5] zisik M.N., Heat Conduction, 2nd ed., John Wiley and Sons, New York, 1993, Appendix 1, pp. 657660. [5 6] Bejan, A., Convection Heat Transfer, 2nd ed ., John Wiley & Sons, New York, 1995, Appendix C, pp. 595602. PAGE 84 70 Appendices PAGE 85 71 A ppendix A: Q Basic Heat Transfer Code of a Wavy Fin Analysi s 'MUTASIM AN ROSARIO Research Basic program' 'this program solves the ordinary differential equations' 'For the radial fin assembly heat transfer with dehumification.' 'X is the adimensional radius variable for fin portion.' 'Y is the derivative of F for fin portion.' 'F is the adimensional temperature for fin portion.' DIM X(1 TO 100), Y(1 TO 100), F(1 TO 100) 'X1 is the adimensional radius variable for wall portion.' 'Y1 is the derivative of F1 for wall portion.' 'F1 is the adimensional temperature for portion.' DIM X1(1 TO 100), Y1(1 TO 100), F1(1 TO 100) 'T is the temperature used to calculate ratio of sensible to total heat.' 'pws saturation pressure.' 'pw partial pressure ofwater vapor .' 'w humidity ratio.' 'the procedure used to calculate the humidity ratio taken from ASHRAE.' DIM T(1 TO 100), pws(1 TO 100), pw(1 TO 100), w(1 TO 100), CON(1 TO 100), R(1 TO 100) 'initial guess 0.01 Sept 14 97 HR = .50 cpa = .24 rma = .075 hfg = 1076 T1 = 32 T2 = 75.2 PAGE 86 72 Appendix A: (Continued) C8 = 1.04404 10000 C9 = 1.129465 10 C10 = 2.702235 / 100 C11 = 1.289036 / 100000 C12 = 2.478068 / 1000000000 C13 = 6.545967 CON2 = C8 / (T2 + 460) + C9 + C10 (T2 + 460) + C11 ((T2 + 460) ^ 2) + C12 ((T2 + 460) ^ 3) + C13 LOG(T2 + 460) pws2 = EXP(CON2) pw2 = HR pws2 w2 = .62198 pw2 / (14.7 pw2) '"INPUT PARAMETERS"' BI1 = 1 BI2 = .1 "PLEASE INPUT STEP SIZE H "; H for the fin calculation' H = .1 'INPUT "PLEASE INPUT DIMENSION OF Y AND F ND "; ND for fin calculation' ND = 11 'INPUT "PLEASE INPUT NUMBER OF STEPS N "; N for fin calculation' N = 11 "N SHOULD BE LESS THAN OR EQUAL TO DIMENSION OF Y AND F"' INPUT CONVERGENCE CRITERION EPS "; EPS' EDGE = H N + 1.5 P RINT "EDGE =", EDGE PARAMETERS' 'INPUT "PLEASE INPUT P"; P' PAGE 87 73 Appendix A: (Continued) P = .25 'INPUT "PLEASE INPUT THETA";THETA THETA = 3.1416 / 4 'INPUT "PLEASE INPUT STEP SIZE H1 "; H1 for wall calculation' H1 = .05 'INPUT "PLEASE INPUT DIMENSION OF Y1 AND F1 ND1 "; ND1 for wall' ND1 = 11 'INPUT "PLEASE INPUT NUMBER OF STEPS N1 "; N1 for wall' N1 = 11 'PRINT "N1 SHOULD BE LESS THAN OR EQUAL TO DIMENSION OF Y1 AND F1"' 'INPUT "PLEASE INPUT K' K = 1 BI = BI2 P / K B = (BI / ((P ^ 2))) EDGE1 = H1 N1 + 1 PRINT "EDGE1 =", EDGE1 'SET INITIAL CONDITIONS' 100 INPUT "INITIAL GUESS `FOR Y1(1)= A1"; A1 PRINT A1 IF (A1 < 1000) THEN GOTO 100 ELSE END IF 40 Y1(1) = A1 'BOUNDARY CONDITION' F1(1) = 1 + (Y1(1) / BI1) PAGE 88 74 Appendix A: (Continued) X1(1) = 1 X(1) = 1.5 FOR IT = 1 TO 2 FOR I = 1 TO (N1 1) RK11 = H1 Y1(1) RK21 = H1 (Y1(I) + (RK11 / 2)) RK31 = H1 (Y1(I) + (RK21 / 2)) RK41 = H1 (Y1(I) + RK31) F1(I + 1) = F1(I) + (RK11 + 2 RK21 + 2 RK31 + RK41) / 6 RK1P1 = H1 0 RK2P1 = H1 ((RK1P1 / 2) / (H1 / 2)) RK3P1 = H1 ((RK2P1 / 2) / (H1 / 2)) RK4P1 = H1 ((RK3P1) / (H1)) Y1(I + 1) = Y1(I) + (RK1P1 + 2 RK2P1 + 2 RK3P1 + RK4P1) / 6 X1(I + 1) = X1(I) + H1 NEXT I 'to calculate Rb at base' I = N1 T(I) = (T1 T2) F1(I) + T2 CON(I) = C8 / (T(I) + 460) + C9 + C10 (T(I) + 460) + C11 ((T(I) + 460) ^ 2) + C12 ((T(I) + 460) ^ 3) + C13 LOG((T(I) + 460)) pws(I) = EXP(CON(I)) pw(I) = HR pws(I) w(I) = .62198 pw(I) / (14.7 pw(I)) R(I) = 1 / (1 + (1 hfg (1 / (T2 T1)) (w2 w(I)) / ((F1(I) cpa)))) PAGE 89 75 Appendix A: (Continued) F(1) = F1(N1) 'BOUNDARY CONDITION' Y(1) = (Y1(N1) + (BI2 / R(N1)) (1 P) F1(N1)) / (K P) FOR I = 1 TO (N 1) RK1 = H Y(I) RK2 = H (Y(I) + (RK1 / 2)) RK3 = H (Y(I) + (RK2 / 2)) RK4 = H (Y(I) + RK3) F(I + 1) = F(I) + (RK1 + 2 RK2 + 2 RK3 + RK4) / 6 T(I) = (T1 T2) F(I) + T2 CON(I) = C8 / (T(I) + 460) + C9 + C10 (T(I) + 460) + C11 ((T(I) + 460) ^ 2) + C12 ((T(I) + 460) ^ 3) + C13 LOG((T(I) + 460)) pws(I) = EXP(CON(I)) pw(I) = HR pws(I) w(I) = .62198 pw(I) / (14.7 pw(I)) R(I) = 1 / (1 + (1 hfg (1 / (T2 T1)) (w2 w(I)) / (F(I) cpa))) RK1P = H (B COS(THETA) F(I) / R(I)) T(I) = (T1 T2) (F(I) + RK1 / 2) + T2 CON(I) = C8 / (T(I) + 460) + C9 + C10 (T(I) + 460) + C11 ((T(I) + 460) ^ 2) + C12 ((T(I) + 460) ^ 3) + C13 LOG((T(I) + 460)) pws(I) = EXP(CON(I)) pw(I) = HR pws(I) w(I) = .62198 pw(I) / (14.7 pw(I)) PAGE 90 76 Appendix A: (Continued) R(I) = 1 / (1 + (1 hfg (1 / (T2 T1)) (w2 w(I)) / ((F(I) + RK1 / 2) cpa))) RK2P = H (B COS(THETA) (F(I) + RK1 / 2) / R(I) + (RK1P / 2) / (H / 2)) T(I) = (T1 T2) (F(I) + RK2 / 2) + T2 CON(I) = C8 / (T(I) + 460) + C9 + C10 (T(I) + 460) + C11 ((T(I) + 460) ^ 2) + C12 ((T(I) + 460) ^ 3) + C13 LOG((T(I) + 460)) pws(I) = EXP(CON(I)) pw(I) = HR pws(I) w(I) = .62198 pw(I) / (14.7 pw(I)) R(I) = 1 / (1 + (1 hfg (1 / (T2 T1)) (w2 w(I)) / ((F(I) + RK2 / 2) cpa))) RK3P = H (B COS(THETA) (F(I) + RK2 / 2) / R(I) + (RK2P / 2) / (H / 2)) CON(I) = C8 / (T(I) + 460) + C9 + C10 (T(I) + 460) + C11 ((T(I) + 460) ^ 2) + C12 ((T(I) + 460) ^ 3) + C13 LOG((T(I) + 460)) pws(I) = EXP(CON(I)) pw(I) = HR pws(I) w(I) = .62198 pw(I) / (14.7 pw(I)) R(I) = 1 / (1 + (1 hfg (1 / (T2 T1)) (w2 w(I)) / ((F(I) + RK3) cpa))) RK4P = H (B cos(THETA) (F(I) + RK3) / R(I) + (RK3P) / (H)) Y(I + 1) = Y(I) + (RK1P + 2 RK2P + 2 RK3P + RK4P) / 6 X(I + 1) = X(I) + H NEXT I PAGE 91 77 Appendix A: (Continued) IF (IT = 1) THEN S1 = Y(N) DA1 = A1 / 50000 A1 = A1 + DA1 Y1(1) = A1 ELSE S2 = Y(N) END IF NEXT IT T(N) = (T1 T2) (F(N)) + T2 CON(N) = C8 / (T(N) + 460) + C9 + C10 (T(N) + 460) + C11 ((T(N) + 460) ^ 2) + C12 ((T(N) + 460) ^ 3) + C13 LOG((T(N) + 460)) pws(N) = EXP(CON(N)) pw(N) = HR pws(N) w(N) = .62198 pw(N) / (14.7 pw(N)) R(N) = 1 / (1 + (1 hfg (1 / (T2 T1)) (w2 w(N)) / ((F(N)) cpa))) 'BOUNDARY CONDITION' YEND = (BI F(N)) / (R(N) P) PRINT "YEND=", YEND S12 = (S2 S1) / DA1 IF (S12 = 0) THEN GOTO 50 ELSE END IF A1 = Y1(1) + (YEND Y(N)) / S12 PAGE 92 78 Appendix A: (Continued) IF (A1 < 1000) THEN PRINT "TRY ANOTHER GUESS FOR A1" GOTO 100 ELSE END IF IF (ABS(Z A1) < EPS) THEN GOTO 50 ELSE END IF Z = A1 GOTO 40 50 X(1) = 1.5 PRINT "T1="; T1; "T2="; T2; "HR="; HR FOR I = 1 TO N1 STEP 1 PRINT I, X1(I), Y1(I), F1(I) NEXT I FOR I = 1 TO N STEP 2 PRINT I, X(I), Y(I), F(I) NEXT I END PAGE 93 79 Appendix B: FIDAP Code for Analysis of Heat Transfer by Jet Impingement B.1 Using Copper "Cu"at Re = 100 FIGEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1, TOLE = 0.0001 ) /POINTS POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 3.1125, Y = 0 ) POINT( ADD, COOR, X = 3.3125, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0.3 ) POINT( ADD, COOR, X = 3.3125, Y = 0.3 ) POINT( ADD, COOR, X = 5.0653, Y = 4.2497 ) POINT( ADD, COOR, X = 9.315, Y = 6.01 ) POINT( ADD, COOR, X = 9.315, Y = 6.21 ) POINT( ADD, COOR, X = 4.9239, Y = 4.3911 ) POINT( ADD, COOR, X = 3.1125, Y = 0.3 ) POINT( ADD, COOR, X = 0, Y = 6.21 ) POINT( ADD, COOR, X = 0, Y = 0.3 ) /LINES POINT ( SELE, ID ) 1 6 CURVE( ADD, LINE ) POINT( SELE, ID ) 6 8 CURVE( ADD, ARC ) POINT ( SELE, ID ) PAGE 94 80 Appendix B: (Continued) 8 9 CURVE( ADD, LINE ) POINT( SELE, ID ) 9 11 CURVE( ADD, ARC ) POINT ( SELE, ID ) 11 2 CURVE( ADD, LINE ) POINT ( SELE, ID ) 11 6 CURVE( ADD, LINE ) POINT ( SELE, ID ) 6 3 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 9 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 13 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 PAGE 95 81 Appendix B: (Continued) 11 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 1 CURVE( ADD, LINE ) /SURFACE POINT ( SELE, ID ) 1 4 12 9 SURFACE ( ADD, POIN, ROWW = 2, NOAD ) //MESH EDGES CURVE( SELE,ID = 1 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 2 ) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 3 ) MEDGE( ADD, SUCC, INTE = 80, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 4 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 5 ) MEDGE( ADD, SUCC, INTE = 80, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 6 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 7 ) PAGE 96 82 Appendix B: (Continued) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 8 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 9 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 10 ) MEDGE( ADD, SUCC, INTE = 9, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 11 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 12 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 13 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 1.1, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 14 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 15 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) /LOOP 1 CURVE( SELE, ID ) 1 9 14 15 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 2 CURVE( SELE, ID ) PAGE 97 83 Appendix B: (Continued) 14 8 12 13 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 3 CURVE( SELE, ID ) 10 6 7 8 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 4 CURVE( SELE, ID ) 2 11 10 9 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 5 CURVE( SELE, ID ) 3 4 5 PAGE 98 84 Appendix B: (Continued) 11 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 1 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 2 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 3 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 4 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 5 ) MFACE( ADD ) // MESHING MFACE( SELE,ID ) 1 2 PAGE 99 85 Appendix B: (Continued) ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "Cu" ) MFACE( SELE,ID ) 3 4 5 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, ENTI ="water" ) /MESH MAP ELEMENT ID ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( SELE,ID = 4 ) MEDGE( MESH, MAP, ENTI = "inlet" ) MEDGE( SELE,ID = 7 ) MEDGE( MESH, MAP, ENTI = "outlet" ) MEDGE( SELE,ID = 5 ) MEDGE( MESH, MAP, ENTI = "surf1" ) MEDGE( SELE,ID = 6 ) MEDGE( MESH, MAP, ENTI = "surf2" ) MEDGE( SELE,ID ) 13 15 MEDGE( MESH, MAP, ENTI = "bottom" ) MEDGE( SELE,ID ) 1 2 3 MEDGE( MESH, MAP, ENTI = "axis" ) PAGE 100 86 Appendix B: (Continued) MEDGE( SELE,ID = 12 ) MEDGE( MESH, MAP, ENTI = "sides" ) MEDGE( SELE,ID ) 8 9 MEDGE( MESH, MAP, ENTI = "interface" ) END( ) FIPREP( ) //Fluid and solid properties /WATER PROPERTIES DENSITY( ADD, SET = "water", CONS = 0.996 ) CONDUCTIVITY( ADD, SET = "water", CONS = 0.0014699 ) VISCOSITY( ADD, SET = "water", CONS = 0.00798 ) SPECIFICHEAT( ADD, SET = "water", CONS = 0.998137 ) SURFACETENSION( ADD, SET = "water", CONS = 73 ) /CU PROPERTIES DENSITY( SET = "Cu", CONS = 8.954 ) CONDUCTIVITY( SET = "Cu", CONS = 0.922562 ) SPECIFICHEAT( SET = "Cu", CONS = 0.0915019 ) ENTITY( ADD, NAME = "Cu", SOLI, PROP = "Cu" ) ENTITY( ADD, NAME = "water", FLUI, PROP = "water" ) ENTITY( ADD, NAME = "inlet", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "surf1", PLOT ) ENTITY( ADD, NAME = "surf2", PLOT ) ENTITY( ADD, NAME = "bottom", PLOT ) ENTITY( ADD, NAME = "axis", PLOT ) PAGE 101 87 Appendix B: (Continued) ENTITY( ADD, NAME = "sides", PLOT ) ENTITY( ADD, NAME = "interface", PLOT, ATTA = "Cu", NATT = "water" ) BODYFORCE( ADD, CONS, FX = 981, FY = 0, FZ = 0 ) PRESSURE( ADD, MIXE = 1e11, DISC ) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE, BOUN ) OPTIONS (ADD, UPWI ) UPWINDING (ADD, STRE ) /You can try different ones to see which one works RELAXATION( ) 0.3, 0.3, 0.3, 0, 0.05, 0.25, 0.25 /0.6, 0.6, 0.6, 0, 0.3, 0.3, 0.3 /0.5, 0.5, 0.5, 0, 0.75, 0.75, 0.75 BCNODE( ADD, URC, ENTI = "axis", ZERO ) BCNODE( ADD, URC, ENTI = "inlet", ZERO ) BCNODE( ADD, UZC, ENTI = "inlet", CONS = 50 ) BCNODE( ADD, TEMP, ENTI = "inlet", CONS = 37 ) BCNODE( ADD, VELO, ENTI = "surf1", ZERO ) BCNODE( ADD, VELO, ENTI = "surf2", ZERO ) BCNODE( ADD, VELO, ENTI = "sides", ZERO ) BCNODE( ADD, VELO, ENTI = "bottom", ZERO ) BCFLUX( ADD, HEAT, ENTI = "bottom", CONS = 5.971 ) BCNODE( ADD, VELO, ENTI = "interface", ZERO ) BCNODE( ADD, VELO, ENTI = "Cu", ZERO ) /ICNODE( VELO, STOKES ) PAGE 102 88 Appendix B: (Continued) /PROBLEM DEFINITION PROBLEM( ADD, 2D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, S.S. = 1500, VELC = 1e5, RESC = 1e5 ) CLIPPING( ADD, MINI ) 0, 0, 0, 0, 37, 0 END( ) CREATE( FISO ) RUN( FISOLV, BACK, AT = "", TIME = "NOW", COMP ) B.2 Using Copper "Cu"at Re = 750 FIGEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1, TOLE = 0.0001 ) /POINTS POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 3.1125, Y = 0 ) POINT( ADD, COOR, X = 3.3125, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0.3 ) POINT( ADD, COOR, X = 3.3125, Y = 0.3 ) POINT( ADD, COOR, X = 5.0653, Y = 4.2497 ) POINT( ADD, COOR, X = 9.315, Y = 6.01 ) POINT( ADD, COOR, X = 9.315, Y = 6.21 ) POINT( ADD, COOR, X = 4.9239, Y = 4.3911 ) POINT( ADD, COOR, X = 3.1125, Y = 0.3 ) PAGE 103 89 Appendix B: (Continued) POINT( ADD, COOR, X = 0, Y = 6.21 ) POINT( ADD, COOR, X = 0, Y = 0.3 ) /LINES POINT ( SELE, ID ) 1 6 CURVE( ADD, LINE ) POINT( SELE, ID ) 6 8 CURVE( ADD, ARC ) POINT ( SELE, ID ) 8 9 CURVE( ADD, LINE ) POINT( SELE, ID ) 9 11 CURVE( ADD, ARC ) POINT ( SELE, ID ) 11 2 CURVE( ADD, LINE ) POINT ( SELE, ID ) 11 6 CURVE( ADD, LINE ) POINT ( SELE, ID ) 6 3 CURVE( ADD, LINE ) PAGE 104 90 Appendix B: (Continued) POINT ( SELE, ID ) 12 9 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 13 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 11 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 1 CURVE( ADD, LINE ) /SURFACE POINT ( SELE, ID ) 1 4 12 9 SURFACE ( ADD, POIN, ROWW = 2, NOAD ) //MESH EDGES CURVE( SELE,ID = 1 ) MEDGE( ADD, SUCC, INTE = 20, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 2 ) PAGE 105 91 Appendix B: (Continued) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 3 ) MEDGE( ADD, SUCC, INTE = 100, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 4 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 5 ) MEDGE( ADD, SUCC, INTE = 100, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 6 ) MEDGE( ADD, SUCC, INTE = 200, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 7 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 8 ) MEDGE( ADD, SUCC, INTE = 200, RATI = 0, 2RAT = 0, PCEN = 0 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 10 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 11 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 12 ) MEDGE( ADD, SUCC, INTE = 20, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 13 ) MEDGE( ADD, SUCC, INTE = 200, RATI = 1.1, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 14 ) MEDGE( ADD, SUCC, INTE = 20, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 15 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 9 ) PAGE 106 92 Appendix B: (Continued) /LOOP 1 CURVE( SELE, ID ) 1 9 14 15 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 2 CURVE( SELE, ID ) 14 8 12 13 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 3 CURVE( SELE, ID ) 10 6 7 8 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 4 CURVE( SELE, ID ) 2 PAGE 107 93 Appendix B: (Continued) 11 10 9 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 5 CURVE( SELE, ID ) 3 4 5 11 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 1 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 2 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 3 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) PAGE 108 94 Appendix B: (Continued) MLOOP( SELE, ID = 4 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 5 ) MFACE( ADD ) // MESHING MFACE( SELE,ID ) 1 2 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "Cu" ) MFACE( SELE,ID ) 3 4 5 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, ENTI ="water" ) /MESH MAP ELEMENT ID ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( SELE,ID = 4 ) MEDGE( MESH, MAP, ENTI = "inlet" ) MEDGE( SELE,ID = 7 ) MEDGE( MESH, MAP, ENTI = "outlet" ) MEDGE( SELE,ID ) 5 6 PAGE 109 95 Appendix B: (Continued) MEDGE( MESH, MAP, ENTI = "surface" ) MEDGE( SELE,ID ) 13 15 MEDGE( MESH, MAP, ENTI = "bottom" ) MEDGE( SELE,ID ) 1 2 3 MEDGE( MESH, MAP, ENTI = "axis" ) MEDGE( SELE,ID = 12 ) MEDGE( MESH, MAP, ENTI = "sides" ) MEDGE( SELE,ID ) 8 9 MEDGE( MESH, MAP, ENTI = "interface" ) END( ) FIPREP( ) //Fluid and solid properties /WATER PROPERTIES DENSITY( ADD, SET = "water", CONS = 0.996 ) CONDUCTIVITY( ADD, SET = "water", CONS = 0.0014699 ) VISCOSITY( ADD, SET = "water", CONS = 0.00798 ) SPECIFICHEAT( ADD, SET = "water", CONS = 0.998137 ) SURFACETENSION( ADD, SET = "water", CONS = 73 ) /SILICON PROPERTIES /DENSITY( ADD, SET = "silicon", CONS = 2.33 ) PAGE 110 96 Appendix B: (Continued) /CONDUCTIVITY( ADD, SET = "silicon", CONS = 0.334608 ) /SPECIFICHEAT( ADD, SET = "silicon", CONS = 0.17006 ) /CU PROPERTIES DENSITY( SET = "Cu", CONS = 8.954 ) CONDUCTIVITY( SET = "Cu", CONS = 0.922562 ) SPECIFICHEAT( SET = "Cu", CONS = 0.0915019 ) ENTITY( ADD, NAME = "Cu", SOLI, PROP = "Cu" ) ENTITY( ADD, NAME = "water", FLUI, PROP = "water" ) ENTITY( ADD, NAME = "inlet", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "surface", PLOT ) ENTITY( ADD, NAME = "bottom", PLOT ) ENTITY( ADD, NAME = "axis", PLOT ) ENTITY( ADD, NAME = "sides", PLOT ) ENTITY( ADD, NAME = "interface", PLOT, ATTA = "Cu", NATT = "water" ) BODYFORCE( ADD, CONS, FX = 981, FY = 0, FZ = 0 ) PRESSURE( ADD, MIXE = 1e11, DISC ) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE, BOUN ) OPTIONS (ADD, UPWI ) UPWINDING (ADD, STRE ) /You can try different ones to see which one works RELAXATION( ) 0.3, 0.3, 0.3, 0, 0.05, 0.25, 0.25 /0.6, 0.6, 0.6, 0, 0.3, 0.3, 0.3 PAGE 111 97 Appendix B: (Continued) /0.5, 0.5, 0.5, 0, 0.75, 0.75, 0.75 BCNODE( ADD, URC, ENTI = "axis", ZERO ) BCNODE( ADD, URC, ENTI = "inlet", ZERO ) BCNODE( ADD, UZC, ENTI = "inlet", CONS = 50 ) BCNODE( ADD, TEMP, ENTI = "inlet", CONS = 37 ) BCNODE( ADD, VELO, ENTI = "surface", ZERO ) /BCNODE( ADD, VELO, ENTI = "surf2", ZERO ) BCNODE( ADD, VELO, ENTI = "sides", ZERO ) BCNODE( ADD, VELO, ENTI = "bottom", ZERO ) BCFLUX( ADD, HEAT, ENTI = "bottom", CONS = 5.971 ) BCNODE( ADD, VELO, ENTI = "interface", ZERO ) BCNODE( ADD, VELO, ENTI = "Cu", ZERO ) /ICNODE( VELO, STOKES ) /PROBLEM DEFINITION PROBLEM( ADD, 2D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, S.S. = 1500, VELC = 1e5, RESC = 1e5 ) CLIPPING( ADD, MINI ) 0, 0, 0, 0, 37, 0 END( ) CREATE( FISO ) RUN( FISOLV, BACK, AT = "", TIME = "NOW", COMP ) B.3 Using Silicon "Si" EXAMPLE 1 FIGEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, PAGE 112 98 Appendix B: (Continued) MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1, TOLE = 0.0001 ) /POINTS POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 3.1125, Y = 0 ) POINT( ADD, COOR, X = 3.3125, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0.3 ) POINT( ADD, COOR, X = 3.3125, Y = 0.3 ) POINT( ADD, COOR, X = 5.0653, Y = 4.2497 ) POINT( ADD, COOR, X = 9.315, Y = 6.01 ) POINT( ADD, COOR, X = 9.315, Y = 6.21 ) POINT( ADD, COOR, X = 4.9239, Y = 4.3911 ) POINT( ADD, COOR, X = 3.1125, Y = 0.3 ) POINT( ADD, COOR, X = 0, Y = 6.21 ) POINT( ADD, COOR, X = 0, Y = 0.3 ) /LINES POINT ( SELE, ID ) 1 6 CURVE( ADD, LINE ) POINT( SELE, ID ) 6 8 CURVE( ADD, ARC ) POINT ( SELE, ID ) 8 9 CURVE( ADD, LINE ) POINT( SELE, ID ) PAGE 113 99 Appendix B: (Continued) 9 11 CURVE( ADD, ARC ) POINT ( SELE, ID ) 11 2 CURVE( ADD, LINE ) POINT ( SELE, ID ) 11 6 CURVE( ADD, LINE ) POINT ( SELE, ID ) 6 3 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 9 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 13 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 11 CURVE( ADD, LINE ) POINT ( SELE, ID ) PAGE 114 100 Appendix B: (Continued) 13 1 CURVE( ADD, LINE ) /SURFACE POINT ( SELE, ID ) 1 4 12 9 SURFACE ( ADD, POIN, ROWW = 2, NOAD ) //MESH EDGES CURVE( SELE,ID = 1 ) MEDGE( ADD, SUCC, INTE = 40, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 2 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 3 ) MEDGE( ADD, SUCC, INTE = 80, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 4 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 5 ) MEDGE( ADD, SUCC, INTE = 80, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 6 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 7 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 8 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 0, 2RAT = 0, PCEN = 0 ) PAGE 115 101 Appendix B: (Continued) CURVE( SELE,ID = 9 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 10 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 11 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 12 ) MEDGE( ADD, SUCC, INTE = 40, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 13 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 1.1, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 14 ) MEDGE( ADD, SUCC, INTE = 40, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 15 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) /LOOP 1 CURVE( SELE, ID ) 1 9 14 15 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 2 CURVE( SELE, ID ) 14 8 12 PAGE 116 102 Appendix B: (Continued) 13 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 3 CURVE( SELE, ID ) 10 6 7 8 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 4 CURVE( SELE, ID ) 2 11 10 9 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 5 CURVE( SELE, ID ) 3 4 5 11 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) PAGE 117 103 Appendix B: (Continued) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 1 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 2 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 3 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 4 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 5 ) MFACE( ADD ) // MESHING MFACE( SELE,ID ) 1 2 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "Cu" ) MFACE( SELE,ID ) PAGE 118 104 Appendix B: (Continued) 3 4 5 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, ENTI ="water" ) /MESH MAP ELEMENT ID ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( SELE,ID = 4 ) MEDGE( MESH, MAP, ENTI = "inlet" ) MEDGE( SELE,ID = 7 ) MEDGE( MESH, MAP, ENTI = "outlet" ) MEDGE( SELE,ID = 5 ) MEDGE( MESH, MAP, ENTI = "surf1" ) MEDGE( SELE,ID = 6 ) MEDGE( MESH, MAP, ENTI = "surf2" ) MEDGE( SELE,ID ) 13 15 MEDGE( MESH, MAP, ENTI = "bottom" ) MEDGE( SELE,ID ) 1 2 3 MEDGE( MESH, MAP, ENTI = "axis" ) MEDGE( SELE,ID = 12 ) MEDGE( MESH, MAP, ENTI = "sides" ) MEDGE( SELE,ID ) PAGE 119 105 Appendix B: (Continued) 8 9 MEDGE( MESH, MAP, ENTI = "interface" ) END( ) FIPREP( ) //Fluid and solid properties /WATER PROPERTIES DENSITY( ADD, SET = "water", CONS = 0.996 ) CONDUCTIVITY( ADD, SET = "water", CONS = 0.0014699 ) VISCOSITY( ADD, SET = "water", CONS = 0.00798 ) SPECIFICHEAT( ADD, SET = "water", CONS = 0.998137 ) SURFACETENSION( ADD, SET = "water", CONS = 73 ) /SILICON PROPERTIES /DENSITY( ADD, SET = "silicon", CONS = 2.33 ) /CONDUCTIVITY( ADD, SET = "silicon", CONS = 0.334608 ) /SPECIFICHEAT( ADD, SET = "silicon", CONS = 0.17006 ) /CU PROPERTIES DENSITY( SET = "Cu", CONS = 8.954 ) CONDUCTIVITY( SET = "Cu", CONS = 0.922562 ) SPECIFICHEAT( SET = "Cu", CONS = 0.0915019 ) ENTITY( ADD, NAME = "Cu", SOLI, PROP = "Cu" ) ENTITY( ADD, NAME = "water", FLUI, PROP = "water" ) ENTITY( ADD, NAME = "inlet", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "surf1", PLOT ) ENTITY( ADD, NAME = "surf2", PLOT ) ENTITY( ADD, NAME = "bottom", PLOT ) PAGE 120 106 Appendix B: (Continued) ENTITY( ADD, NAME = "axis", PLOT ) ENTITY( ADD, NAME = "sides", PLOT ) ENTITY( ADD, NAME = "interface", PLOT, ATTA = "Cu", NATT = "water" ) BODYFORCE( ADD, CONS, FX = 981, FY = 0, FZ = 0 ) PRESSURE( ADD, MIXE = 1e11, DISC ) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE, BOUN ) OPTIONS (ADD, UPWI ) UPWINDING (ADD, STRE ) /You can try different ones to see which one works RELAXATION( ) 0.3, 0.3, 0.3, 0, 0.05, 0.25, 0.25 /0.6, 0.6, 0.6, 0, 0.3, 0.3, 0.3 /0.5, 0.5, 0.5, 0, 0.75, 0.75, 0.75 BCNODE( ADD, URC, ENTI = "axis", ZERO ) BCNODE( ADD, URC, ENTI = "inlet", ZERO ) BCNODE( ADD, UZC, ENTI = "inlet", CONS = 13.35341 ) BCNODE( ADD, TEMP, ENTI = "inlet", CONS = 37 ) BCNODE( ADD, VELO, ENTI = "surf1", ZERO ) BCNODE( ADD, VELO, ENTI = "surf2", ZERO ) BCNODE( ADD, VELO, ENTI = "sides", ZERO ) BCNODE( ADD, VELO, ENTI = "bottom", ZERO ) BCFLUX( ADD, HEAT, ENTI = "bottom", CONS = 2.9855 ) BCNODE( ADD, VELO, ENTI = "interface", ZERO ) BCNODE( ADD, VELO, ENTI = "Cu", ZERO ) PAGE 121 107 Appendix B: (Continued) /ICNODE( VELO, STOKES ) /PROBLEM DEFINITION PROBLEM( ADD, 2D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, S.S. = 1500, VELC = 1e5, RESC = 1e5 ) CLIPPING( ADD, MINI ) 0, 0, 0, 0, 37, 0 END( ) CREATE( FISO ) RUN( FISOLV, BACK, AT = "", TIME = "NOW", COMP ) B .4 Using Titanium "CuNi" EXAMPLE 1 FIGEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1, TOLE = 0.0001 ) /POINTS POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 3.1125, Y = 0 ) POINT( ADD, COOR, X = 3.3125, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0 ) POINT( ADD, COOR, X = 9.315, Y = 0.3 ) POINT( ADD, COOR, X = 3.3125, Y = 0.3 ) POINT( ADD, COOR, X = 5.0653, Y = 4.2497 ) POINT( ADD, COOR, X = 9.315, Y = 6.01 ) POINT( ADD, COOR, X = 9.315, Y = 6.21 ) PAGE 122 108 A ppendix B: (Continued) POINT( ADD, COOR, X = 4.9239, Y = 4.3911 ) POINT( ADD, COOR, X = 3.1125, Y = 0.3 ) POINT( ADD, COOR, X = 0, Y = 6.21 ) POINT( ADD, COOR, X = 0, Y = 0.3 ) /LINES POINT ( SELE, ID ) 1 6 CURVE( ADD, LINE ) POINT( SELE, ID ) 6 8 CURVE( ADD, ARC ) POINT ( SELE, ID ) 8 9 CURVE( ADD, LINE ) POINT( SELE, ID ) 9 11 CURVE( ADD, ARC ) POINT ( SELE, ID ) 11 2 CURVE( ADD, LINE ) POINT ( SELE, ID ) 11 6 CURVE( ADD, LINE ) POINT ( SELE, ID ) 6 PAGE 123 109 Appendix B: (Continued) 3 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 9 CURVE( ADD, LINE ) POINT ( SELE, ID ) 12 13 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 11 CURVE( ADD, LINE ) POINT ( SELE, ID ) 13 1 CURVE( ADD, LINE ) /SURFACE POINT ( SELE, ID ) 1 4 12 9 SURFACE ( ADD, POIN, ROWW = 2, NOAD ) //MESH EDGES PAGE 124 110 Appendix B: (Continued) CURVE( SELE,ID = 1 ) MEDGE( ADD, SUCC, INTE = 40, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 2 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 3 ) MEDGE( ADD, SUCC, INTE = 80, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 4 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 5 ) MEDGE( ADD, SUCC, INTE = 80, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 6 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 7 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 8 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 9 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 10 ) MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 11 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 12 ) MEDGE( ADD, SUCC, INTE = 40, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 13 ) MEDGE( ADD, SUCC, INTE = 140, RATI = 1.1, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 14 ) PAGE 125 111 Appendix B: (Continued) MEDGE( ADD, SUCC, INTE = 40, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE,ID = 15 ) MEDGE( ADD, SUCC, INTE = 10, RATI = 0, 2RAT = 0, PCEN = 0 ) /LOOP 1 CURVE( SELE, ID ) 1 9 14 15 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 2 CURVE( SELE, ID ) 14 8 12 13 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 3 CURVE( SELE, ID ) 10 6 7 8 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) PAGE 126 112 Appendix B: (Continued) /LOOP 4 CURVE( SELE, ID ) 2 11 10 9 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) /LOOP 5 CURVE( SELE, ID ) 3 4 5 11 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 1 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 2 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 3 ) PAGE 127 113 Appendix B: (Continued) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 4 ) MFACE( ADD ) //ADDING MESH FACE SURFACE( SELE, ID = 1 ) MLOOP( SELE, ID = 5 ) MFACE( ADD ) // MESHING MFACE( SELE,ID ) 1 2 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "CuNi" ) MFACE( SELE,ID ) 3 4 5 ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, ENTI ="water" ) /MESH MAP ELEMENT ID ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( SELE,ID = 4 ) MEDGE( MESH, MAP, ENTI = "inlet" ) MEDGE( SELE,ID = 7 ) MEDGE( MESH, MAP, ENTI = "outlet" ) PAGE 128 114 Appendix B: (Continued) MEDGE( SELE,ID = 5 ) MEDGE( MESH, MAP, ENTI = "surf1" ) MEDGE( SELE,ID = 6 ) MEDGE( MESH, MAP, ENTI = "surf2" ) MEDGE( SELE,ID ) 13 15 MEDGE( MESH, MAP, ENTI = "bottom" ) MEDGE( SELE,ID ) 1 2 3 MEDGE( MESH, MAP, ENTI = "axis" ) MEDGE( SELE,ID = 12 ) MEDGE( MESH, MAP, ENTI = "sides" ) MEDGE( SELE,ID ) 8 9 MEDGE( MESH, MAP, ENTI = "interface" ) END( ) FIPREP( ) //Fluid and solid properties /WATER PROPERTIES DENSITY( ADD, SET = "water", CONS = 0.996 ) CONDUCTIVITY( ADD, SET = "water", CONS = 0.0014699 ) VISCOSITY( ADD, SET = "water", CONS = 0.00798 ) SPECIFICHEAT( ADD, SET = "water", CONS = 0.998137 ) PAGE 129 115 Appendix B: (Continued) SURFACETENSION( ADD, SET = "water", CONS = 73 ) /Constantan (CuNi)PROPERTIES DENSITY( ADD, SET = "CuNi", CONS = 8.9 ) CONDUCTIVITY( ADD, SET = "CuNi", CONS = 0.04657497 ) SPECIFICHEAT( ADD, SET = "CuNi", CONS = 0.39 ) /SILICON PROPERTIES /DENSITY( ADD, SET = "silicon", CONS = 2.33 ) /CONDUCTIVITY( ADD, SET = "silicon", CONS = 0.334608 ) /SPECIFICHEAT( ADD, SET = "silicon", CONS = 0.17006 ) /CU PROPERTIES /DENSITY( SET = "Cu", CONS = 8.954 ) /CONDUCTIVITY( SET = "Cu", CONS = 0.922562 ) /SPECIFICHEAT( SET = "Cu", CONS = 0.0915019 ) ENTITY( ADD, NAME = "CuNi", SOLI, PROP = "CuNi" ) ENTITY( ADD, NAME = "water", FLUI, PROP = "water" ) ENTITY( ADD, NAME = "inlet", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "surf1", PLOT ) ENTITY( ADD, NAME = "surf2", PLOT ) ENTITY( ADD, NAME = "bottom", PLOT ) ENTITY( ADD, NAME = "axis", PLOT ) ENTITY( ADD, NAME = "sides", PLOT ) ENTITY( ADD, NAME = "interface", PLOT, ATTA = "CuNi", NATT = "water" ) BODYFORCE( ADD, CONS, FX = 981, FY = 0, FZ = 0 ) PRESSURE( ADD, MIXE = 1e11, DISC ) DATAPRINT( ADD, CONT ) PAGE 130 116 Appendix B: (Continued) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE, BOUN ) OPTIONS (ADD, UPWI ) UPWINDING (ADD, STRE ) /You can try different ones to see which one works RELAXATION( ) 0.3, 0.3, 0.3, 0, 0.05, 0.25, 0.25 /0.6, 0.6, 0.6, 0, 0.3, 0.3, 0.3 /0.5, 0.5, 0.5, 0, 0.75, 0.75, 0.75 BCNODE( ADD, URC, ENTI = "axis", ZERO ) BCNODE( ADD, URC, ENTI = "inlet", ZERO ) BCNODE( ADD, UZC, ENTI = "inlet", CONS = 13.35341 ) BCNODE( ADD, TEMP, ENTI = "inlet", CONS = 37 ) BCNODE( ADD, VELO, ENTI = "surf1", ZERO ) BCNODE( ADD, VELO, ENTI = "surf2", ZERO ) BCNODE( ADD, VELO, ENTI = "sides", ZERO ) BCNODE( ADD, VELO, ENTI = "bottom", ZERO ) BCFLUX( ADD, HEAT, ENTI = "bottom", CONS = 2.9855 ) BCNODE( ADD, VELO, ENTI = "interface", ZERO ) BCNODE( ADD, VELO, ENTI = "CuNi", ZERO ) /ICNODE( VELO, STOKES ) /PROBLEM DEFINITION PROBLEM( ADD, 2D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, S.S. = 1500, VELC = 1e5, RESC = 1e5 ) CLIPPING( ADD, MINI ) 0, 0, 0, 0, 37, 0 PAGE 131 117 Appendix B: (Continued) END( ) CREATE( FISO ) RUN( FISOLV, BACK, AT = "", TIME = "NOW", COMP ) xml version 1.0 encoding UTF8 standalone no record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam 22 Ka 4500 controlfield tag 007 crbnuuuuuu 008 s2011 flu ob 000 0 eng d datafield ind1 8 ind2 024 subfield code a E14SFE0004827 035 (OCoLC) 040 FHM c FHM 049 FHMM 090 XX9999 (Online) 1 100 Elsheikh, Mutasim Mohamed Sarour. 0 245 Numerical simulations of heat transfer processes in a dehumidifying wavy fin and a confined liquid jet impingement on various surfaces h [electronic resource] / by Mutasim Mohamed Sarour Elsheikh. 260 [Tampa, Fla] : b University of South Florida, 2011. 500 Title from PDF of title page. Document formatted into pages; contains 131 pages. 502 Thesis (M.S.M.E.)University of South Florida, 2011. 504 Includes bibliographical references. 516 Text (Electronic thesis) in PDF format. 520 ABSTRACT: This thesis consists of two different research problems. In the first one, the heat transfer characteristic of wavy fin assembly with dehumidification is carried out. In general, fin tube heat exchangers are employed in a wide variety of engineering applications, such as cooling coils for air conditioning, air preheaters in power plants and for heat dissipation from engine coolants in automobile radiators. In these heat exchangers, a heat transfer fluid such as water, oil, or refrigerant, flows through a parallel tube bank, while a second heat transfer fluid, such as air, is directed across the tubes. Since the principal resistance is much greater on the air side than on the tube side, enhanced surfaces in the form of wavy fins are used in aircooled heat exchangers to improve the overall heat transfer performance. In heating, ventilation, and air conditioning systems (HVAC), the air stream is cooled and dehumidified as it passes through the cooling coils, circulating the refrigerant. Heat and mass transfer take place when the coil surface temperature in most cooling coils is below the dew point temperature of the air being cooled. This thesis presents a simplified analysis of combined heat and mass transfer in wavyfinned cooling coils by considering condensing water film resistance for a fully wet fin in dehumidifier coil operation during air condition. The effects of variation of the cold fluid temperature (5˚C 5˚C), air side temperature (25˚C 35˚C), and relative humidity (50% 70%) on the dimensionless temperature distribution and the augmentation factor are investigated and compared with those under dry conditions. In addition, comparison of the wavy fin with straight radial or rectangular fin under the same conditions were investigated and the results show that the wavy fin has better heat dissipation because of the greater area. The results demonstrate that the overall fin efficiency is dependent on the relative humidity of the surrounding air and the total surface area of the fin. In addition, the findings of the present work are in good agreement with experimental data. The second problem investigated is the heat transfer analysis of confined liquid jet impingement on various surfaces. The objective of this computational study is to characterize the convective heat transfer of a confined liquid jet impinging on a curved surface of a solid body, while the body is being supplied with a uniform heat flux at its opposite flat surface. Both convex and concave configurations of the curved surface are investigated. The confinement plate has the same shape as the curved surface. Calculations were done for various solid materials, namely copper, aluminum, Constantan, and silicon; at twodimensional jet. For this research, Reynolds numbers ranging from 750 to 2000 for various nozzle widths channel spacing, radii of curvature, and base thicknesses of the solid body, were used. Results are presented in terms of dimensionless solidfluid interface temperature, heat transfer coefficient, and local and average Nusselt numbers. The increments of Reynolds numbers increase local Nusselt numbers over the entire solidfluid interface. Decreasing the nozzle width, channel spacing, plate thickness or curved surface radius of curvature all enhanced the local Nusselt number. Results show that a convex surface is more effective compared to a flat or concave surface. Numerical simulation results are validated by comparing them with experimental data for flat and concave surfaces. 538 Mode of access: World Wide Web. System requirements: World Wide Web browser and PDF reader. 590 Advisor: Rahman, Muhammad M. 653 Conjugates Heat Transfer Fullyconfined Fluid Jet Impingement Heat Flux Steady State Transient Analysis 690 Dissertations, Academic z USF x Mechanical Engineering Masters. 773 t USF Electronic Theses and Dissertations. 4 856 u http://digital.lib.usf.edu/?e14.4827 